text: noisicerP noisicerP noisicerP 1 1 1 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 Multi−resolution HPM(HELEN68) , AP: 88.40% Multi−resolution HPM(HELEN68) , AP: 79.04% Multi−resolution HPM(HELEN68) , AP: 97.23% 0.3 HPM(HELEN68) , AP: 87.72% 0.3 HPM(HELEN68) , AP: 77.60% 0.3 HPM(HELEN68) , AP: 97.65% Independent occlusion(HELEN68) , AP: 87.84% Independent occlusion(HELEN68) , AP: 77.58% Independent occlusion(HELEN68) , AP: 97.66% HPM−occ(HELEN68) , AP: 84.11% 0.2 HPM−occ(HELEN68) , AP: 70.82% 0.2 HPM−occ(HELEN68) , AP: 97.40% 0.2 DPM(HELEN68) , AP: 82.47% DPM(HELEN68) , AP: 67.47% DPM(HELEN68) , AP: 97.24% DPM(MULTIPE68) , AP: 77.81% 0.1 DPM(MULTIPE68) , AP: 59.26% 0.1 DPM(MULTIPE68) , AP: 94.55% 0.1 Cascade DPM(AFLW21) , AP: 77.61% Cascade DPM(AFLW21) , AP: 60.50% Cascade DPM(AFLW21) , AP: 96.69% 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Recall Recall Recall (a) UCI-OFD (b) UCI-OFD occluded (c) UCI-OFD visible Figure 10: Precision-Recall curves of face detection on our UCI-OFD dataset (a) for all of the faces, (b) occluded subset and (c) visible subset. On the visible subset our model, DPM retrained on HELEN68 and Cascade DPM [41] have almost similar performances, but our model signiﬁcantly outperforms these methods on the occluded subset and it has a better overall performance. Cascade DPM uses many accelerate techniques, which may reject some of the faces. Its maximum recall for the visible faces is near 100%, while its maximum recall for the occluded faces is only 60%. The initial drop in the Precision-Recall of this method for the occluded subset is because its returned bounding boxes for some of the high scored occluded faces are not accurate and do not have the minimum 0.5 overlap with the ground-truth bounding boxes. outperforms many such methods on occluded faces (Occluded object parts using positive evidence of coherent occlusion. HELEN, COFW). In comparing pose regression and part-based models, there Acknowledgements: This work was supported in part by NSF seem to be several interesting trade-offs. In our experiments, grants IIS-1253538 and DBI-1262547. we see a general trend in which error distribution curves for References pose regression and part-based models cross, suggesting that pose regression yields very accurate localization for a subset of [1] V. Blanz, T. Vetter, Face recognition based on ﬁtting a 3d morphable model, PAMI 25 (9) (2003) 1063–1074. images relative to the HPM but fails for some other proportion [2] A. Martinez, S. Du, A model of the perception of facial expressions even at very large error thresholds. Unlike pose regression, the of emotion by humans: Research overview and perspectives, JMLR 13 part model performs detection, eliminating the need for detec- (2012) 1589–1608. tion as a pre-process and improving robustness. In particular, [3] X. Zhu, D. Ramanan, Face detection, pose estimation, and landmark lo- calization in the wild, in: CVPR, 2012, pp. 2879–2886. we are able to detect many heavily occluded faces which would [4] G. Ghiasi, C. Fowlkes, Occlusion coherence: Localizing occluded faces not be detected by a standard cascade detector and hence in- with a hierarchical deformable part model, in: CVPR, 2014, pp. 1899– accessible to pose regression. We ﬁnd that the HPM tends to 1906. generalize well across datasets suggesting it avoids some over- [5] V. Jain, E. Learned-Miller, Fddb: A benchmark for face detection in un- constrained settings, Tech. Rep. UM-CS-2010-009, University of Mas- ﬁtting problems present in pose regression. sachusetts, Amherst (2010). This ﬂexibility currently comes with a computational cost. [6] A. L. Yuille, P. W. Hallinan, D. S. Cohen, Feature extraction from faces The run-time of our model implementation built on dynamic using deformable templates, IJCV 8 (2) (1992) 99–111. programming lags signiﬁcantly behind those of regression-based, [7] T. F. Cootes, G. J. Edwards, C. J. Taylor, Active appearance models, ∼ PAMI 23 (6) (2001) 681–685. feed-forward approaches. Our implementation takes 10s to [8] I. Matthews, S. Baker, Active appearance models revisited, IJCV 60 (2) run on a typical COFW image, roughly 100x slower than RCPR (2004) 135–164. or DCNN based approaches. However, the HPM is amenable [9] S. Milborrow, F. Nicolls, Locating facial features with an extended active shape model, in: ECCV, 2008, pp. 504–513. to implementation on a GPU which may address most of this [10] L. Wiskott, J.-M. Fellous, N. Kuiger, C. Von Der Malsburg, Face recog- runtime gap. nition by elastic bunch graph matching, PAMI 19 (7) (1997) 775–779. Finally, we note several avenues for future work. Perfor- [11] L. Gu, T. Kanade, 3d alignment of face in a single image, in: CVPR, mance depends on the graphical independence structure of the 2006, pp. 1305–1312. [12] M. Valstar, B. Martinez, X. Binefa, M. Pantic, Facial point detection using model which should ideally be learned from data. While our boosted regression and graph models, in: CVPR, 2010, pp. 2729–2736. model implicitly represents the pattern of part occlusions, it [13] B. Efraty, C. Huang, S. K. Shah, I. A. Kakadiaris, Facial landmark detec- does not integrate local image evidence for the occluder itself. tion in uncontrolled conditions, in: IJCB, 2011, pp. 1–8. A natural extension would be to add local ﬁlters that detect the [14] P. N. Belhumeur, D. W. Jacobs, D. J. Kriegman, N. Kumar, Localizing parts of faces using a consensus of exemplars, in: CVPR, 2011, pp. 545– presence of an occluding contour between the occluded and 552. non-occluded landmarks. Such ﬁlters could be shared across [15] X. P. Burgos-Artizzu, P. Perona, P. Dolla´r, Robust face landmark estima- parts to avoid increasing too much the overall computation cost tion under occlusion, in: ICCV, 2013, pp. 1513–1520. while moving closer to our goal of explaining away missing [16] X. Cao, Y. Wei, F. Wen, J. Sun, Face alignment by explicit shape regres- sion, in: CVPR, 2012, pp. 2887–2894. 12 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: noisicerP

text: noisicerP

text: noisicerP

text: Multi−resolution

text: AP:

text: Multi−resolution

text: AP:

text: Multi−resolution

text: AP:

text: AP:

text: AP:

text: AP:

text: Independent

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text: AP:

text: AP:

text: AP:

text: AP:

text: AP:

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text: Cascade

text: AP:

text: Cascade

text: AP:

text: Cascade

text: AP:

text: Recall

text: Recall

text: Recall

text: (a)

text: UCI-OFD

text: (b)

text: UCI-OFD

text: occluded

text: (c)

text: UCI-OFD

text: visible

text: Figure

text: Precision-Recall

text: curves

text: of

text: face

text: detection

text: on

text: our

text: UCI-OFD

text: dataset

text: (a)

text: for

text: all

text: of

text: the

text: faces,

text: (b)

text: occluded

text: subset

text: and

text: (c)

text: visible

text: subset.

text: On

text: the

text: visible

text: subset

text: our

text: model,

text: DPM

text: retrained

text: on

text: and

text: Cascade

text: DPM

text: have

text: almost

text: similar

text: performances,

text: but

text: our

text: model

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text: outperforms

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text: methods

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text: the

text: occluded

text: subset

text: and

text: it

text: has

text: a

text: better

text: overall

text: performance.

text: Cascade

text: DPM

text: uses

text: many

text: accelerate

text: techniques,

text: which

text: may

text: reject

text: some

text: of

text: the

text: faces.

text: Its

text: maximum

text: recall

text: for

text: the

text: visible

text: faces

text: is

text: near

text: while

text: its

text: maximum

text: recall

text: for

text: the

text: occluded

text: faces

text: is

text: only

text: The

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text: drop

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text: the

text: Precision-Recall

text: of

text: this

text: method

text: for

text: the

text: occluded

text: subset

text: is

text: because

text: its

text: returned

text: bounding

text: boxes

text: for

text: some

text: of

text: the

text: high

text: scored

text: occluded

text: faces

text: are

text: not

text: accurate

text: and

text: do

text: not

text: have

text: the

text: minimum

text: overlap

text: with

text: the

text: ground-truth

text: bounding

text: boxes.

text: outperforms

text: many

text: such

text: methods

text: on

text: occluded

text: faces

text: (Occluded

text: object

text: parts

text: using

text: positive

text: evidence

text: of

text: coherent

text: occlusion.

text: HELEN,

text: COFW).

text: In

text: comparing

text: pose

text: regression

text: and

text: part-based

text: models,

text: there

text: Acknowledgements:

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: NSF

text: seem

text: to

text: be

text: several

text: interesting

text: trade-offs.

text: In

text: our

text: experiments,

text: grants

text: and

text: we

text: see

text: a

text: general

text: trend

text: in

text: which

text: error

text: distribution

text: curves

text: for

text: References

text: pose

text: regression

text: and

text: part-based

text: models

text: cross,

text: suggesting

text: that

text: pose

text: regression

text: yields

text: very

text: accurate

text: localization

text: for

text: a

text: subset

text: of

text: V.

text: Blanz,

text: T.

text: Vetter,

text: Face

text: recognition

text: based

text: on

text: ﬁtting

text: a

text: morphable

text: model,

text: PAMI

text: images

text: relative

text: to

text: the

text: HPM

text: but

text: fails

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text: other

text: proportion

text: A.

text: Martinez,

text: S.

text: Du,

text: A

text: model

text: of

text: the

text: perception

text: of

text: facial

text: expressions

text: even

text: at

text: very

text: large

text: error

text: thresholds.

text: Unlike

text: pose

text: regression,

text: the

text: of

text: emotion

text: by

text: humans:

text: Research

text: overview

text: and

text: perspectives,

text: JMLR

text: part

text: model

text: performs

text: detection,

text: eliminating

text: the

text: need

text: for

text: detec-

text: tion

text: as

text: a

text: pre-process

text: and

text: improving

text: robustness.

text: In

text: particular,

text: X.

text: Zhu,

text: D.

text: Ramanan,

text: Face

text: detection,

text: pose

text: estimation,

text: and

text: landmark

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text: calization

text: in

text: the

text: wild,

text: in:

text: CVPR,

text: pp.

text: we

text: are

text: able

text: to

text: detect

text: many

text: heavily

text: occluded

text: faces

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text: would

text: G.

text: Ghiasi,

text: C.

text: Fowlkes,

text: Occlusion

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text: detected

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text: model,

text: in:

text: CVPR,

text: pp.

text: accessible

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text: pose

text: regression.

text: We

text: ﬁnd

text: that

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text: HPM

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text: over-

text: V.

text: Jain,

text: E.

text: Learned-Miller,

text: Fddb:

text: A

text: benchmark

text: for

text: face

text: detection

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text: un-

text: constrained

text: settings,

text: Tech.

text: Rep.

text: University

text: of

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text: ﬁtting

text: problems

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text: pose

text: regression.

text: sachusetts,

text: Amherst

text: This

text: ﬂexibility

text: currently

text: comes

text: with

text: a

text: computational

text: cost.

text: A.

text: L.

text: Yuille,

text: P.

text: W.

text: Hallinan,

text: D.

text: S.

text: Cohen,

text: Feature

text: extraction

text: from

text: faces

text: The

text: run-time

text: of

text: our

text: model

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text: built

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text: templates,

text: IJCV

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text: T.

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text: G.

text: J.

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text: C.

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text: Active

text: appearance

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text: PAMI

text: feed-forward

text: approaches.

text: Our

text: implementation

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text: to

text: I.

text: Matthews,

text: S.

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text: Active

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text: IJCV

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text: typical

text: COFW

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text: F.

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text: Locating

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text: features

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text: model,

text: in:

text: ECCV,

text: pp.

text: to

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text: L.

text: Wiskott,

text: J.-M.

text: Fellous,

text: N.

text: Kuiger,

text: C.

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text: Face

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text: runtime

text: gap.

text: nition

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text: matching,

text: PAMI

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text: Kanade,

text: alignment

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text: face

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text: the

text: pp.

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text: X.

text: Binefa,

text: M.

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text: Facial

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text: detection

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text: data.

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text: our

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text: models,

text: in:

text: CVPR,

text: pp.

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text: implicitly

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text: the

text: pattern

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text: part

text: occlusions,

text: it

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text: C.

text: Huang,

text: S.

text: K.

text: Shah,

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text: A.

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text: Facial

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text: N.

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text: pp.

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text: rorre egarevA LFPW (29) COFW (29) 0.55 HPM model training dataset SR AE SR AE 0.5 DPM RCPR-occ LFPW29 88.95 0.073 63.44 0.115 RCPR RCPR-occ LFPW29+ 98.95 0.038 63.64 0.096 0.45 RCPR retrain RCPR-occ COFW29 89.01 0.071 76.28 0.091 0.4 RCPR COFW29 91.05 0.064 79.25 0.085 HPM LFPW68,INR- 97.37 0.050 86.76 0.075 0.35 HPM HELEN68,INR- 98.42 0.049 90.71 0.072 0.3 HPM HELEN68,PAS- 98.95 0.048 92.09 0.070 0.25 0.2 Table 2: We ﬁnd HPM generalizes well across datasets while pose regression has a strong dependence on training data. Localization performance is mea- 0.15 sured by success rate (SR) and average error (AE). The RCPR model trained 0.1 on COFW performs much better on COFW test data compared to RCPR-occ trained on LFPW29+ (79% SR vs 64% SR) but has much worse performance 0.05 on LFPW test data compared to that model (91% SR vs 99% SR). Good perfor- 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Overlap ratio mance on LFPW also depends heavily on including additional warped positive instances (LFPW29+ vs LFPW29). The HPM trained on LFPW68 has high success rates on both COFW (87%) and LFPW (%97) test data. Last two rows of the table show the performance of HPM when a different training data set Figure 7: We analyze the landmark localization average error of RCPR, HPM (HELEN68) is used for training. This dataset has more variation and more im- and DPM for different overlap ratio with the ground-truth face boxes. For ages (1758) compared to LFPW68 (682) and improves performance of HPM on RCPR we change the minimum overlap ratio of the initial bounding boxes and both test datasets. Training on more negative images (6000 images from PAS- the ground-truth face boxes. For HPM and DPM, we change the minimum over- CAL) decreases localization error of our model compared to using only INRIA lap threshold of the returned detections and ground-truth boxes. RCPR is very negatives. sensitive to the amount of overlap and its performance decreases rapidly as the overlap ratio decreases. But, HPM and DPM are robust to the overlap threshold and they can maintain the same performance over different thresholds. TCDN [22] and CFSS [20]. Since these methods operate in the general framework of pose regression, performing iterative on the original COFW 29-landmark test set [15] consisting of reﬁnement of predicted landmark locations, they are sensitive 507 internet photos depicting a wide variety of more difﬁcult to initial bounding box location. We tested both models us- poses and includes a signiﬁcant amount of occlusion. Since ing the standardized detection bounding boxes provided by the COFW training only contains 29 landmarks (we only performed 300-W benchmark [36] rather than tight cropping images to additional annotations on test data), we evaluated models trained the ground-truth landmark locations. We used the pre-trained on LFPW68 and HELEN68. Fig. 6(c) shows that HPM achieves TCDCN model available online while for CFSS we retrained a signiﬁcantly lower average error than RCPR and higher suc- the model using the standard detector bounding boxes. In both cess rates for all but the smallest (< 0.06) localization success cases, average error was signiﬁcantly worse than previously re- thresholds. ported results, highlighting the sensitivity of these methods to initialization. Localization Results (COFW68). We tested our model trained on LPFW68 and HELEN68 training data on this benchmark Dependence of Localization on Detection. A key beneﬁt of the and compared with CFSS, TCDCN and RCPR-occ (Fig. 6 HPM (and DPM [3]) approach is that the same model serves to (c)). For CFSS and TCDCN we used the publicly available both detect and localize the landmarks. In contrast, pose regres- pre-trained models which were trained on HELEN68, LFPW68 sion methods such as RCPR, TCDN or CFSS require that the and AFW68 (TCDCN is also pretrained on MAFL dataset). For face already be detected. This distinction becomes particularly RCPR-occ we used the authors’ code to train a model on HE- important for occluded faces since detection is signiﬁcantly less LEN68 and LFPW68 training sets. Note we that couldn’t train accurate (see Detection experiments below). the full RCPR 68-landmark model with occlusion since HE- To highlight the dependence of landmark localization on LEN68 and LFPW68 do not have occlusion and COFW train is accurate detection, we benchmarked average localization error only labeled with 29 landmarks. for varying degrees of overlap between the hypothesized detec- tion and ground-truth bounding box on the COFW test set. As Localization Results (IBUG 68). This dataset contains 68 land- shown in Fig. 7, decreasing the overlap ratio has no affect HPM mark annotations for 135 faces in difﬁcult poses and expres- / DPM performance since there are never false positives in the sion [36]. For testing our method on this dataset, we follow vicinity of the face that score higher than one with high overlap previous work and trained our model on combined HELEN68 ratio. In contrast, RCPR performs signiﬁcantly worse when ini- and LFPW68 training data provided by 300-W. Since IBUG in- tialized from bounding boxes that do not have high overlap with cludes many side view faces we trained a variant of our model the face. Since the area over which RCPR searches is learned with 7 viewpoints. We compare our model with published per- from training data, we also retrained a version of RCPR for formance of several state-of-the-art methods in Table 1 and achieve each degrees of overlap. This yielded improved performance comparable performance. but still shows a signiﬁcant fall off in performance compared In addition to reporting values from the published litera- to the HPM. As noted above, we encountered similar behavior ture, we also re-evaluated two recent top-performing models: 9 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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label: paragraph

font: default

text: egarevA

text: LFPW

label: table

text: COFW

text: HPM

text: model

text: training

text: dataset

text: SR

text: AE

text: SR

text: AE

text: DPM

text: RCPR-occ

text: RCPR

text: RCPR-occ

text: RCPR

text: retrain

text: RCPR-occ

text: RCPR

text: HPM

text: HPM

text: HPM

text: Table

text: We

label: caption

text: ﬁnd

text: HPM

text: generalizes

text: well

text: across

text: datasets

text: while

text: pose

text: regression

text: has

text: a

text: strong

text: dependence

text: on

text: training

text: data.

text: Localization

text: performance

text: is

text: mea-

text: sured

text: by

text: success

text: rate

text: (SR)

text: and

text: average

text: error

text: (AE).

text: The

text: RCPR

text: model

text: trained

text: on

text: COFW

text: performs

text: much

text: better

text: on

text: COFW

text: test

text: data

text: compared

text: to

text: RCPR-occ

text: trained

text: on

text: SR

text: vs

text: SR)

text: but

text: has

text: much

text: worse

text: performance

text: on

text: LFPW

text: test

text: data

text: compared

text: to

text: that

text: model

text: SR

text: vs

text: SR).

text: Good

text: perfor-

text: Overlap

text: ratio

text: mance

text: on

text: LFPW

text: also

text: depends

text: heavily

text: on

text: including

text: additional

text: warped

text: positive

text: instances

text: vs

text: The

text: HPM

text: trained

text: on

text: has

text: high

text: success

text: rates

text: on

text: both

text: COFW

text: and

text: LFPW

text: test

text: data.

text: Last

text: two

text: rows

text: of

text: the

text: table

text: show

text: the

text: performance

text: of

text: HPM

text: when

text: a

text: different

text: training

text: data

text: set

text: Figure

text: We

text: analyze

text: the

text: landmark

text: localization

text: average

text: error

text: of

text: RCPR,

text: HPM

text: is

text: used

text: for

text: training.

text: This

text: dataset

text: has

text: more

text: variation

text: and

text: more

text: im-

text: and

text: DPM

text: for

text: different

text: overlap

text: ratio

text: with

text: the

text: ground-truth

text: face

text: boxes.

text: For

text: ages

text: compared

text: to

text: and

text: improves

text: performance

text: of

text: HPM

text: on

text: RCPR

text: we

text: change

text: the

text: minimum

text: overlap

text: ratio

text: of

text: the

text: initial

text: bounding

text: boxes

text: and

text: both

text: test

text: datasets.

text: Training

text: on

text: more

text: negative

text: images

text: images

text: from

text: PAS-

text: the

text: ground-truth

text: face

text: boxes.

text: For

text: HPM

text: and

text: DPM,

text: we

text: change

text: the

text: minimum

text: over-

text: CAL)

text: decreases

text: localization

text: error

text: of

text: our

text: model

text: compared

text: to

text: using

text: only

text: INRIA

text: lap

text: threshold

text: of

text: the

text: returned

text: detections

text: and

text: ground-truth

text: boxes.

text: RCPR

text: is

text: very

text: negatives.

text: sensitive

text: to

text: the

text: amount

text: of

text: overlap

text: and

text: its

text: performance

text: decreases

text: rapidly

text: as

text: the

text: overlap

text: ratio

text: decreases.

text: But,

text: HPM

text: and

text: DPM

text: are

text: robust

text: to

text: the

text: overlap

text: threshold

text: and

text: they

text: can

text: maintain

text: the

text: same

text: performance

text: over

text: different

text: thresholds.

text: TCDN

text: and

text: CFSS

text: Since

text: these

text: methods

text: operate

text: in

text: the

text: general

text: framework

text: of

text: pose

text: regression,

text: performing

text: iterative

text: on

text: the

text: original

text: COFW

text: test

text: set

text: consisting

text: of

text: reﬁnement

text: of

text: predicted

text: landmark

text: locations,

text: they

text: are

text: sensitive

text: internet

text: photos

text: depicting

text: a

text: wide

text: variety

text: of

text: more

text: difﬁcult

text: to

text: initial

text: bounding

text: box

text: location.

text: We

text: tested

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text: and

text: includes

text: a

text: signiﬁcant

text: amount

text: of

text: occlusion.

text: Since

text: ing

text: the

text: standardized

text: detection

text: bounding

text: boxes

text: provided

text: by

text: the

text: COFW

text: training

text: only

text: contains

text: landmarks

text: (we

text: only

text: performed

text: benchmark

text: rather

text: than

text: tight

text: cropping

text: images

text: to

text: additional

text: annotations

text: on

text: test

text: data),

text: we

text: evaluated

text: models

text: trained

text: the

text: ground-truth

text: landmark

text: locations.

text: We

text: used

text: the

text: pre-trained

text: on

text: and

text: Fig.

text: shows

text: that

text: HPM

text: achieves

text: TCDCN

text: model

text: available

text: online

text: while

text: for

text: CFSS

text: we

text: retrained

text: a

text: signiﬁcantly

text: lower

text: average

text: error

text: than

text: RCPR

text: and

text: higher

text: suc-

text: the

text: model

text: using

text: the

text: standard

text: detector

text: bounding

text: boxes.

text: In

text: both

text: cess

text: rates

text: for

text: all

text: but

text: the

text: smallest

text: localization

text: success

text: cases,

text: average

text: error

text: was

text: signiﬁcantly

text: worse

text: than

text: previously

text: re-

text: thresholds.

text: ported

text: results,

text: highlighting

text: the

text: sensitivity

text: of

text: these

text: methods

text: to

text: initialization.

text: Localization

text: Results

text: We

text: tested

text: our

text: model

text: trained

text: on

text: and

text: training

text: data

text: on

text: this

text: benchmark

text: Dependence

text: of

text: Localization

text: on

text: Detection.

text: A

text: key

text: beneﬁt

text: of

text: the

text: and

text: compared

text: with

text: CFSS,

text: TCDCN

text: and

text: RCPR-occ

text: (Fig.

text: HPM

text: (and

text: DPM

text: approach

text: is

text: that

text: the

text: same

text: model

text: serves

text: to

text: (c)).

text: For

text: CFSS

text: and

text: TCDCN

text: we

text: used

text: the

text: publicly

text: available

text: both

text: detect

text: and

text: localize

text: the

text: landmarks.

text: In

text: contrast,

text: pose

text: regres-

text: pre-trained

text: models

text: which

text: were

text: trained

text: on

text: sion

text: methods

text: such

text: as

text: RCPR,

text: TCDN

text: or

text: CFSS

text: require

text: that

text: the

text: and

text: (TCDCN

text: is

text: also

text: pretrained

text: on

text: MAFL

text: dataset).

text: For

text: face

text: already

text: be

text: detected.

text: This

text: distinction

text: becomes

text: particularly

text: RCPR-occ

text: we

text: used

text: the

text: authors’

text: code

text: to

text: train

text: a

text: model

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text: HE-

text: important

text: for

text: occluded

text: faces

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text: detection

text: is

text: signiﬁcantly

text: less

text: and

text: training

text: sets.

text: Note

text: we

text: that

text: couldn’t

text: train

text: accurate

text: (see

text: Detection

text: experiments

text: below).

text: the

text: full

text: RCPR

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text: occlusion

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text: highlight

text: the

text: dependence

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text: landmark

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text: COFW

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text: accurate

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text: landmarks.

text: for

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text: of

text: overlap

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text: tion

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text: poses

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text: DPM

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text: there

text: are

text: never

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text: the

text: sion

text: For

text: testing

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text: method

text: on

text: this

text: dataset,

text: we

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text: vicinity

text: of

text: the

text: face

text: that

text: score

text: higher

text: than

text: one

text: with

text: high

text: overlap

text: previous

text: work

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text: model

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text: ratio.

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text: RCPR

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text: Since

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text: which

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text: viewpoints.

text: We

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text: our

text: model

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text: training

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text: methods

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text: by a d-vector derived by average pooling, which is quite neat vention of ASR, the input layer involves a window of 20- and efﬁcient, but loses much information of the test signal, dimensional Fbanks. The window size is set to 21, which was such as the distributional property and the temporal constraint. found to be optimal in our work. The DNN structure involves One of the main contribution of this paper is to investigate how 4 hidden layers, and each consists of 200 units. The units of to utilize the temporal constraint in the DNN-based approach. the output layer correspond to the speakers in the training data, The DNN model has been studied in speaker recognition and the number is 80 in our experiment. The 1-hot encoding in several ways. For example, in [10], DNNs trained for scheme is used to label the target, and the training criterion ASR were used to replace the UBM model to derive the is set to cross entropy. The learning rate is set to 0.008 at acoustic statistics for i-vector models. In [11], a DNN was the beginning, and is halved whenever no improvement on a used to replace PLDA to improve discriminative capability of cross-validation (CV) set is found. The training process stops i-vectors. All these methods rely on the generative framework, when the learning rate is too small and the improvement on i.e., the i-vector model. The DNN-based feature learning the CV set is too marginal. Once the DNN has been trained presented in this paper is purely discriminative, without any successfully, the speaker-discriminative features can be read generative model involved. from the last hidden layer. III. DNN-BASED FEATURE LEARNING In the test phase, the features are extracted for all frames of the given utterance. To derive utterance-based representations, It is well-known that DNNs can learn task-oriented fea- an average pooling approach was used in [8], where the frame- tures from raw input layer by layer. This property has been level features are averaged and the resultant vector is used to employed in ASR where phone-discriminative features are represent the speaker. This vector is called ‘d-vector’ in [8], learned from very low-level features such as Fbanks or even and we adopt this name in this work. The same methods used spectra [7]. It has been shown that with a well-trained DNN, for i-vectors can be used for d-vectors to conduct the test, for variations irrelevant to the learning task can be gradually example by computing the cosine distance or the PLDA score. eliminated when the feature propagates through the DNN structure layer by layer. This feature learning is so powerful A. Comparison between i-vectors and d-vectors that in ASR, the primary Fbank feature has defeated the MFCC The two kinds of speaker vectors, the d-vector and the i- feature that was carefully designed by people and dominated vector, are fundamentally different. I-vectors are based on a in ASR for several decades. linear Gaussian model, for which the learning is unsupervised This property can be also employed to learn speaker- and the learning criterion is maximum likelihood on acoustic discriminative features. Actually researchers have put much features; in contrast, d-vectors are based on neural networks, effort in searching for features that are more discriminative for which the learning is supervised, and the learning criterion for speakers [6], but the effort is mostly vain and the MFCC is maximum discrimination for speakers. This difference leads is still the most popular choice. The success of DNNs in ASR to several advantages with d-vectors: First, it is a ‘discrimina- suggests a new direction, that speaker-discriminative features tive’ vector, which represents speakers by removing speaker- can be learned from data instead of being crafted by hand. The irrelevant variance, and so sensitive to speakers and invariant learning can be easily done and the process is rather similar as to other disturbance; second, it is a ‘local’ speaker description in ASR, with the only difference that in speaker recognition, that uses only local context, so can be inferred from very short the learning goal is to discriminate different speakers. utterances; third, it relies on ‘universal’ data to learn the DNN model, which makes it possible to learn from large amounts Fbank of data that are task-independent. (40*21 dim) IV. IMPROVED DEEP FEATURE LEARNING d-vector is the averaged activations from the last hidden layers There are several limitations in the implementation of the feature learning paradigm presented in the previous section. P(spk1) First, it does not involve any prior knowledge in model training, for example phone identities. Second, the simple P(spk ) 2 average pooling does not consider the temporal information which is particularly important for text-dependent recognition tasks. Several approaches are proposed in this section to address these problems. A. Phone-dependent training A potential problem of the DNN-based feature learning P(spk ) N described in the previous section is that it is a ‘blind learning’, i.e., the features are learned from raw data without any prior knowledge. This means that the learning purely relies on Fully-connected maxout hidden layers. the complex deep structure of the DNN model and a large amount of data to discover speaker-discriminative patterns. Output layer is removed in If the training data is abundant, this is often not a problem; enrollment and evaluation. however in tasks with a limited amount of data, for instance Fig. 1. The DNN structure used for learning speaker-discriminative features. the text-dependent task in our hand, this blind learning tends to be difﬁcult because there are too many speaker-irrelevant Fig. 1 presents the DNN structure used in this work for variations involved in the raw data, particularly phone con- speaker-discriminative feature learning. Following the con- tents. ##LTFigure## ##LTLine##

text: by

text: a

text: d-vector

text: derived

text: by

text: average

text: pooling,

text: which

text: is

text: quite

text: neat

text: vention

text: of

text: ASR,

text: the

text: input

text: layer

text: involves

text: a

text: window

text: of

text: and

text: efﬁcient,

text: but

text: loses

text: much

text: information

text: of

text: the

text: test

text: signal,

text: dimensional

text: Fbanks.

text: The

text: window

text: size

text: is

text: set

text: to

text: which

text: was

text: such

text: as

text: the

text: distributional

text: property

text: and

text: the

text: temporal

text: constraint.

text: found

text: to

text: be

text: optimal

text: in

text: our

text: work.

text: The

text: DNN

text: structure

text: involves

text: One

text: of

text: the

text: main

text: contribution

text: of

text: this

text: paper

text: is

text: to

text: investigate

text: how

text: hidden

text: layers,

text: and

text: each

text: consists

text: of

text: units.

text: The

text: units

text: of

text: to

text: utilize

text: the

text: temporal

text: constraint

text: in

text: the

text: DNN-based

text: approach.

text: the

text: output

text: layer

text: correspond

text: to

text: the

text: speakers

text: in

text: the

text: training

text: data,

text: The

text: DNN

text: model

text: has

text: been

text: studied

text: in

text: speaker

text: recognition

text: and

text: the

text: number

text: is

text: in

text: our

text: experiment.

text: The

text: encoding

text: in

text: several

text: ways.

text: For

text: example,

text: in

text: DNNs

text: trained

text: for

text: scheme

text: is

text: used

text: to

text: label

text: the

text: target,

text: and

text: the

text: training

text: criterion

text: ASR

text: were

text: used

text: to

text: replace

text: the

text: UBM

text: model

text: to

text: derive

text: the

text: is

text: set

text: to

text: cross

text: entropy.

text: The

text: learning

text: rate

text: is

text: set

text: to

text: at

text: acoustic

text: statistics

text: for

text: i-vector

text: models.

text: In

text: a

text: DNN

text: was

text: the

text: beginning,

text: and

text: is

text: halved

text: whenever

text: no

text: improvement

text: on

text: a

text: used

text: to

text: replace

text: PLDA

text: to

text: improve

text: discriminative

text: capability

text: of

text: cross-validation

text: (CV)

text: set

text: is

text: found.

text: The

text: training

text: process

text: stops

text: i-vectors.

text: All

text: these

text: methods

text: rely

text: on

text: the

text: generative

text: framework,

text: when

text: the

text: learning

text: rate

text: is

text: too

text: small

text: and

text: the

text: improvement

text: on

text: i.e.,

text: the

text: i-vector

text: model.

text: The

text: DNN-based

text: feature

text: learning

text: the

text: CV

text: set

text: is

text: too

text: marginal.

text: Once

text: the

text: DNN

text: has

text: been

text: trained

text: presented

text: in

text: this

text: paper

text: is

text: purely

text: discriminative,

text: without

text: any

text: successfully,

text: the

text: speaker-discriminative

text: features

text: can

text: be

text: read

text: generative

text: model

text: involved.

text: from

text: the

text: last

text: hidden

text: layer.

text: III.

text: DNN-BASED

text: FEATURE

text: LEARNING

text: In

text: the

text: test

text: phase,

text: the

text: features

text: are

text: extracted

text: for

text: all

text: frames

text: of

text: the

text: given

text: utterance.

text: To

text: derive

text: utterance-based

text: representations,

text: It

text: is

text: well-known

text: that

text: DNNs

text: can

text: learn

text: task-oriented

text: fea-

text: an

text: average

text: pooling

text: approach

text: was

text: used

text: in

text: where

text: the

text: frame-

text: tures

text: from

text: raw

text: input

text: layer

text: by

text: layer.

text: This

text: property

text: has

text: been

text: level

text: features

text: are

text: averaged

text: and

text: the

text: resultant

text: vector

text: is

text: used

text: to

text: employed

text: in

text: ASR

text: where

text: phone-discriminative

text: features

text: are

text: represent

text: the

text: speaker.

text: This

text: vector

text: is

text: called

text: ‘d-vector’

text: in

text: learned

text: from

text: very

text: low-level

text: features

text: such

text: as

text: Fbanks

text: or

text: even

text: and

text: we

text: adopt

text: this

text: name

text: in

text: this

text: work.

text: The

text: same

text: methods

text: used

text: spectra

text: It

text: has

text: been

text: shown

text: that

text: with

text: a

text: well-trained

text: DNN,

text: for

text: i-vectors

text: can

text: be

text: used

text: for

text: d-vectors

text: to

text: conduct

text: the

text: test,

text: for

text: variations

text: irrelevant

text: to

text: the

text: learning

text: task

text: can

text: be

text: gradually

text: example

text: by

text: computing

text: the

text: cosine

text: distance

text: or

text: the

text: PLDA

text: score.

text: eliminated

text: when

text: the

text: feature

text: propagates

text: through

text: the

text: DNN

text: structure

text: layer

text: by

text: layer.

text: This

text: feature

text: learning

text: is

text: so

text: powerful

text: A.

text: Comparison

text: between

text: i-vectors

text: and

text: d-vectors

text: that

text: in

text: ASR,

text: the

text: primary

text: Fbank

text: feature

text: has

text: defeated

text: the

text: MFCC

text: The

text: two

text: kinds

text: of

text: speaker

text: vectors,

text: the

text: d-vector

text: and

text: the

text: i-

text: feature

text: that

text: was

text: carefully

text: designed

text: by

text: people

text: and

text: dominated

text: vector,

text: are

text: fundamentally

text: different.

text: I-vectors

text: are

text: based

text: on

text: a

text: in

text: ASR

text: for

text: several

text: decades.

text: linear

text: Gaussian

text: model,

text: for

text: which

text: the

text: learning

text: is

text: unsupervised

text: This

text: property

text: can

text: be

text: also

text: employed

text: to

text: learn

text: speaker-

text: and

text: the

text: learning

text: criterion

text: is

text: maximum

text: likelihood

text: on

text: acoustic

text: discriminative

text: features.

text: Actually

text: researchers

text: have

text: put

text: much

text: features;

text: in

text: contrast,

text: d-vectors

text: are

text: based

text: on

text: neural

text: networks,

text: effort

text: in

text: searching

text: for

text: features

text: that

text: are

text: more

text: discriminative

text: for

text: which

text: the

text: learning

text: is

text: supervised,

text: and

text: the

text: learning

text: criterion

text: for

text: speakers

text: but

text: the

text: effort

text: is

text: mostly

text: vain

text: and

text: the

text: MFCC

text: is

text: maximum

text: discrimination

text: for

text: speakers.

text: This

text: difference

text: leads

text: is

text: still

text: the

text: most

text: popular

text: choice.

text: The

text: success

text: of

text: DNNs

text: in

text: ASR

text: to

text: several

text: advantages

text: with

text: d-vectors:

text: First,

text: it

text: is

text: a

text: ‘discrimina-

text: suggests

text: a

text: new

text: direction,

text: that

text: speaker-discriminative

text: features

text: tive’

text: vector,

text: which

text: represents

text: speakers

text: by

text: removing

text: speaker-

text: can

text: be

text: learned

text: from

text: data

text: instead

text: of

text: being

text: crafted

text: by

text: hand.

text: The

text: irrelevant

text: variance,

text: and

text: so

text: sensitive

text: to

text: speakers

text: and

text: invariant

text: learning

text: can

text: be

text: easily

text: done

text: and

text: the

text: process

text: is

text: rather

text: similar

text: as

text: to

text: other

text: disturbance;

text: second,

text: it

text: is

text: a

text: ‘local’

text: speaker

text: description

text: in

text: ASR,

text: with

text: the

text: only

text: difference

text: that

text: in

text: speaker

text: recognition,

text: that

text: uses

text: only

text: local

text: context,

text: so

text: can

text: be

text: inferred

text: from

text: very

text: short

text: the

text: learning

text: goal

text: is

text: to

text: discriminate

text: different

text: speakers.

text: utterances;

text: third,

text: it

text: relies

text: on

text: ‘universal’

text: data

text: to

text: learn

text: the

text: DNN

text: model,

text: which

text: makes

text: it

text: possible

text: to

text: learn

text: from

text: large

text: amounts

text: Fbank

text: of

text: data

text: that

text: are

text: task-independent.

text: dim)

text: IV.

text: IMPROVED

text: DEEP

text: FEATURE

text: LEARNING

text: d-vector

text: is

text: the

text: averaged

text: activations

text: from

text: the

text: last

text: hidden

text: layers

text: There

text: are

text: several

text: limitations

text: in

text: the

text: implementation

text: of

text: the

text: feature

text: learning

text: paradigm

text: presented

text: in

text: the

text: previous

text: section.

text: First,

text: it

text: does

text: not

text: involve

text: any

text: prior

text: knowledge

text: in

text: model

text: training,

text: for

text: example

text: phone

text: identities.

text: Second,

text: the

text: simple

text: P(spk

text: average

text: pooling

text: does

text: not

text: consider

text: the

text: temporal

text: information

text: which

text: is

text: particularly

text: important

text: for

text: text-dependent

text: recognition

text: tasks.

text: Several

text: approaches

text: are

text: proposed

text: in

text: this

text: section

text: to

text: address

text: these

text: problems.

text: A.

text: Phone-dependent

text: training

text: A

text: potential

text: problem

text: of

text: the

text: DNN-based

text: feature

text: learning

text: P(spk

text: N

text: described

text: in

text: the

text: previous

text: section

text: is

text: that

text: it

text: is

text: a

text: ‘blind

text: learning’,

text: i.e.,

text: the

text: features

text: are

text: learned

text: from

text: raw

text: data

text: without

text: any

text: prior

text: knowledge.

text: This

text: means

text: that

text: the

text: learning

text: purely

text: relies

text: on

text: Fully-connected

text: maxout

text: hidden

text: layers.

text: the

text: complex

text: deep

text: structure

text: of

text: the

text: DNN

text: model

text: and

text: a

text: large

text: amount

text: of

text: data

text: to

text: discover

text: speaker-discriminative

text: patterns.

text: Output

text: layer

text: is

text: removed

text: in

text: If

text: the

text: training

text: data

text: is

text: abundant,

text: this

text: is

text: often

text: not

text: a

text: problem;

text: enrollment

text: and

text: evaluation.

text: however

text: in

text: tasks

text: with

text: a

text: limited

text: amount

text: of

text: data,

text: for

text: instance

text: Fig.

text: The

text: DNN

text: structure

text: used

text: for

text: learning

text: speaker-discriminative

text: features.

text: the

text: text-dependent

text: task

text: in

text: our

text: hand,

text: this

text: blind

text: learning

text: tends

text: to

text: be

text: difﬁcult

text: because

text: there

text: are

text: too

text: many

text: speaker-irrelevant

text: Fig.

text: presents

text: the

text: DNN

text: structure

text: used

text: in

text: this

text: work

text: for

text: variations

text: involved

text: in

text: the

text: raw

text: data,

text: particularly

text: phone

text: con-

text: speaker-discriminative

text: feature

text: learning.

text: Following

text: the

text: con-

text: tents.

text: ##LTFigure##

text: ##LTLine##


text: 14 A. AKHMEDOV AND W. ZHANG no longer trivial. In , it represents a π (Y (1/p , 1/q , · · · , 1/p , 1/q ) \ νT ) 1 g 1 1 g g meridian of . T Since , after the ﬁber sum we have π (E(1) \ (ν(T )) = 1 c = d = 1 1 in the fundamental group of . As a consequence of X (p , q , · · · , p , q ) g 1 1 g g this, we obtain the following presentation for the fundamental group of : X (p , q , · · · , p , q ) g 1 1 g g ap1 = 1, 1 bq1 = 1, 1 · · · , · · · , apg = 1, g bqg = 1, g g Π [a , b ] = 1. j=1 j j By setting , we obtain a symplectic -manifold p = q = · · · = p = q = 1 4 1 1 g g with trivial fundamental group. The symplectic -manifold X (1, 1, · · · , 1, 1) 4 g has the fundamental group , a free group of rank . In gen- X (1, 0, · · · 1, 0) F g g g eral, the symplectic -manifold has the fun- 4 X (1, 0, · · · 1, 0, 1, p , · · · , 1, p ) g,k 1 k damental group (∗n Z)∗(∗k Z ). The manifold X (1, 0, · · · 1, 0, 1, p , · · · , 1, p ) i=1 i=1 pi g,k 1 k has Euler number and signature . Since the ﬁrst Betti number e = 12 σ = −8 − (cid:3) b = g, we have b+ = g + 1 and b = g + 9. 1 Now, we are ready to prove Theorem 1.1. For the convenience of readers, let us rephrase it in the below. Theorem 3.6. For any ﬁnitely presented group which is dual ﬁnite tor- G (cid:48) sion, there is a ﬁnitely presented group with surjective group homomor- G (cid:48) (cid:16) phism , such that there is a minimal symplectic -manifolds G G 4 X(G) (cid:48) with b+(X(G)) = b (G) + 1, π (X(G)) = G and c2(X(G)) = 0. Moreover, (cid:48) 1 1 1 . b (G) = b (G ) 1 1 Proof. Now, let be any ﬁnitely presented G = (cid:104)x , · · · , x | l , · · · , l (cid:105) 1 k 1 m group whose Abelianization is denoted by . To construct manifolds with AG fundamental group , we use similar strategy as for Proposition 3.5. G A new ingredient is the bridge move operation, as illustrated in Figure 1. It is used to resolve the self-intersection points of an immersed loop on surface . Namely, for each self-intersection point with two segments Σ x l and intersecting locally, we increase the genus of by adding a “bridge” m Σ with going across the bridge and going under the bridge. This operation l m of resolving self-intersection points is introduced in [9] (see also [30]).

text: A.

text: AKHMEDOV

text: AND

text: W.

text: ZHANG

text: no

text: longer

text: trivial.

text: In

text: it

text: represents

text: a

text: π

text: (Y

text: νT

text: g

text: g

text: g

text: meridian

text: of

text: T

text: Since

text: after

text: the

text: ﬁber

text: sum

text: we

text: have

text: π

text: (ν(T

text: c

text: d

text: in

text: the

text: fundamental

text: group

text: of

text: As

text: a

text: consequence

text: of

text: X

text: (p

text: q

text: p

text: q

text: g

text: g

text: g

text: this,

text: we

text: obtain

text: the

text: following

text: presentation

text: for

text: the

text: fundamental

text: group

text: of

text: X

text: (p

text: q

text: p

text: q

text: g

text: g

text: g

text: apg

text: g

text: bqg

text: g

text: g

text: Π

text: [a

text: b

text: j

text: j

text: By

text: setting

text: we

text: obtain

text: a

text: symplectic

text: -manifold

text: p

text: q

text: p

text: q

text: g

text: g

text: with

text: trivial

text: fundamental

text: group.

text: The

text: symplectic

text: -manifold

text: X

text: g

text: has

text: the

text: fundamental

text: group

text: a

text: free

text: group

text: of

text: rank

text: In

text: gen-

text: X

text: F

text: g

text: g

text: g

text: eral,

text: the

text: symplectic

text: -manifold

text: has

text: the

text: fun-

text: X

text: p

text: p

text: g,k

text: k

text: damental

text: group

text: (∗n

text: Z)∗(∗k

text: Z

text: The

text: manifold

text: X

text: p

text: p

text: pi

text: g,k

text: k

text: has

text: Euler

text: number

text: and

text: signature

text: Since

text: the

text: ﬁrst

text: Betti

text: number

text: e

text: σ

text: −8

text: b

text: g,

text: we

text: have

text: b+

text: g

text: and

text: b

text: g

text: Now,

text: we

text: are

text: ready

text: to

text: prove

text: Theorem

text: For

text: the

text: convenience

text: of

text: readers,

text: let

text: us

text: rephrase

text: it

text: in

text: the

text: below.

text: Theorem

text: For

text: any

text: ﬁnitely

text: presented

text: group

text: which

text: is

text: dual

text: ﬁnite

text: tor-

text: G

text: sion,

text: there

text: is

text: a

text: ﬁnitely

text: presented

text: group

text: with

text: surjective

text: group

text: homomor-

text: G

text: phism

text: such

text: that

text: there

text: is

text: a

text: minimal

text: symplectic

text: -manifolds

text: G

text: G

text: X(G)

text: with

text: b+(X(G))

text: b

text: (G)

text: π

text: (X(G))

text: G

text: and

text: Moreover,

text: b

text: (G)

text: b

text: (G

text: Proof.

text: Now,

text: let

text: be

text: any

text: ﬁnitely

text: presented

text: G

text: x

text: l

text: l

text: k

text: m

text: group

text: whose

text: Abelianization

text: is

text: denoted

text: by

text: To

text: construct

text: manifolds

text: with

text: AG

text: fundamental

text: group

text: we

text: use

text: similar

text: strategy

text: as

text: for

text: Proposition

text: G

text: A

text: new

text: ingredient

text: is

text: the

text: bridge

text: move

text: operation,

text: as

text: illustrated

text: in

text: Figure

text: It

text: is

text: used

text: to

text: resolve

text: the

text: self-intersection

text: points

text: of

text: an

text: immersed

text: loop

text: on

text: surface

text: Namely,

text: for

text: each

text: self-intersection

text: point

text: with

text: two

text: segments

text: Σ

text: x

text: l

text: and

text: intersecting

text: locally,

text: we

text: increase

text: the

text: genus

text: of

text: by

text: adding

text: a

text: “bridge”

text: m

text: Σ

text: with

text: going

text: across

text: the

text: bridge

text: and

text: going

text: under

text: the

text: bridge.

text: This

text: operation

text: l

text: m

text: of

text: resolving

text: self-intersection

text: points

text: is

text: introduced

text: in

text: (see

text: also


text: 20 A. AKHMEDOV AND W. ZHANG We form the symplectic sum of along the Y (1/p , 1/q , · · · , 1/p , 1/q ) (cid:48)(cid:48) n 1 1 n n genus two surface with . Z (1, 1) (cid:48)(cid:48) X (p , q , · · · , p , q ) = Y (1/p , 1/q , · · · , 1/p , 1/q )# Z (1, 1) n 1 1 n n n 1 1 n n id We claim the fundamental group of is free group X (1, 1, 1, 1, 1, 0 · · · , 1, 0) n F − with n − 2 generators. n 2 From Seifert-Van Kampen theorem, the fundamental group is a quotient of the following group: (cid:48)(cid:48) π (Y (1, 1, 1, 1, 1, 1/0, · · · , 1, 1/0) \ νΣ ) ∗ π (Z (1, 1) \ νΣ¯ ) (9) 1 n 2 1 − (cid:81)2 . (cid:104)a = α , b = α , a = a2, b = α , µ(Σ ) = µ(Σ¯ ) 1(cid:105) 1 1 1 2 2 3 2 4 2 2 All the relations in (5) continue to hold in (9) except possibly for n [c , d ] = j=1 j j . This product may no longer be trivial and now represents a meridian of 1 . Σ 2 − − − − − − − In (9), we have a = [b 1, d 1] = [b 1, [c 1, b ] 1] = [b 1, [b , c 1]]. Since 1 1 1 1 1 2 1 2 1 in (8), we conclude that . Since we also have [α , α ] = 1 [b , b ] = 1 [b , c ] = 2 4 1 2 1 1 from (5), we easily deduce that . From and ﬁrst eight relations 1 a = 1 a = 1 1 1 in (5), we have , , , are all trivial when , . We could also kill a b c d i = 1 2 i i i i all for . Consequently, all the generators would be all c i = 3, · · · , n α i − − − − i killed. Especially, α = [α 1, α 1] = [a 1, b 1] = 1. Finally, since [α , α ] 3 1 4 1 2 3 4 is trivial and hence the generators die as well. In conclusion, the g , · · · , g 1 s fundamental group is a free group generated by . d , · · · , d 3 n In general, we apply bridge moves as we did in the proof of Theorem 3.6. Let whose Q-abelianization equals G = (cid:104)x , · · · , x | l , · · · , l (cid:105) Q. We repr1esent k 1 by m . For any relation in a AG ⊗ x , · · · , x d , · · · , d l 1 k 3 k+2 i given presentation of , we ﬁnd a immersed loop generated by with G γ d i i . After performing bridge moves, the self-intersection points i = 3, · · · , k + 2 (cid:48) of are resolve with the cost of increasing genus to . Similarly, γ k + 2 k i (cid:48) we order the bridges corresponding to the order of . In addition to the γ i Luttinger surgeries in the beginning of , we perform Luttinger 2(k + 2) + 4 surgeries to generators , of added bridges: c d g g (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) (c × b , c , −1), (d × b , d , (−1)c(g) 1), k + 3 ≤ g ≤ k , g 1 g g 2 g (cid:48) and to the embedded loops : γ i (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (γ × b , γ , −1), · · · , (γ × b , γ , −1). 1 2 1 m 2 m We call this manifold Y +(G). Similarly, we apply symplectic sum of (cid:48)(cid:48) Y +(G) along the genus two surface Σ descending from Σ ×pt with Z (1, 1), 2 2 and denote the resulting symplectic manifold by X+(G). Exactly the same argument as in Theorem 3.6, the abelianization of is Q if the presentation π (X(G) AG ⊗ G = (cid:104)x , · · · , x | l , · · · , l (cid:105) 1 1 k 1 m is dual ﬁnite torsion. Moreover, by Usher’s theorem [46], X+(G) is minimal. ##LTLine##

text: A.

text: AKHMEDOV

text: AND

text: W.

text: ZHANG

text: We

text: form

text: the

text: symplectic

text: sum

text: of

text: along

text: the

text: Y

text: n

text: n

text: n

text: genus

text: two

text: surface

text: with

text: Z

text: X

text: (p

text: q

text: p

text: q

text: Y

text: Z

text: n

text: n

text: n

text: n

text: n

text: n

text: id

text: We

text: claim

text: the

text: fundamental

text: group

text: of

text: is

text: free

text: group

text: X

text: n

text: F

text: with

text: n

text: generators.

text: n

text: From

text: Seifert-Van

text: Kampen

text: theorem,

text: the

text: fundamental

text: group

text: is

text: a

text: quotient

text: of

text: the

text: following

text: group:

text: π

text: (Y

text: νΣ

text: π

text: (Z

text: νΣ¯

text: n

text: α

text: b

text: α

text: a

text: b

text: α

text: µ(Σ

text: µ(Σ¯

text: All

text: the

text: relations

text: in

text: continue

text: to

text: hold

text: in

text: except

text: possibly

text: for

text: n

text: [c

text: d

text: j

text: j

text: This

text: product

text: may

text: no

text: longer

text: be

text: trivial

text: and

text: now

text: represents

text: a

text: meridian

text: of

text: Σ

text: In

text: we

text: have

text: a

text: [b

text: d

text: [b

text: [c

text: b

text: [b

text: [b

text: c

text: Since

text: in

text: we

text: conclude

text: that

text: Since

text: we

text: also

text: have

text: [α

text: α

text: [b

text: b

text: [b

text: c

text: from

text: we

text: easily

text: deduce

text: that

text: From

text: and

text: ﬁrst

text: eight

text: relations

text: a

text: a

text: in

text: we

text: have

text: are

text: all

text: trivial

text: when

text: We

text: could

text: also

text: kill

text: a

text: b

text: c

text: d

text: i

text: i

text: i

text: i

text: i

text: all

text: for

text: Consequently,

text: all

text: the

text: generators

text: would

text: be

text: all

text: c

text: i

text: n

text: α

text: i

text: i

text: killed.

text: Especially,

text: α

text: [α

text: α

text: [a

text: b

text: Finally,

text: since

text: [α

text: α

text: is

text: trivial

text: and

text: hence

text: the

text: generators

text: die

text: as

text: well.

text: In

text: conclusion,

text: the

text: g

text: g

text: s

text: fundamental

text: group

text: is

text: a

text: free

text: group

text: generated

text: by

text: d

text: d

text: n

text: In

text: general,

text: we

text: apply

text: bridge

text: moves

text: as

text: we

text: did

text: in

text: the

text: proof

text: of

text: Theorem

text: Let

text: whose

text: Q-abelianization

text: equals

text: G

text: x

text: l

text: l

text: Q.

text: We

text: k

text: by

text: m

text: For

text: any

text: relation

text: in

text: a

text: AG

text: x

text: x

text: d

text: d

text: l

text: k

text: i

text: given

text: presentation

text: of

text: we

text: ﬁnd

text: a

text: immersed

text: loop

text: generated

text: by

text: with

text: G

text: γ

text: d

text: i

text: i

text: After

text: performing

text: bridge

text: moves,

text: the

text: self-intersection

text: points

text: i

text: k

text: of

text: are

text: resolve

text: with

text: the

text: cost

text: of

text: increasing

text: genus

text: to

text: Similarly,

text: γ

text: k

text: k

text: i

text: we

text: order

text: the

text: bridges

text: corresponding

text: to

text: the

text: order

text: of

text: In

text: addition

text: to

text: the

text: γ

text: i

text: Luttinger

text: surgeries

text: in

text: the

text: beginning

text: of

text: we

text: perform

text: Luttinger

text: surgeries

text: to

text: generators

text: of

text: added

text: bridges:

text: c

text: d

text: g

text: g

text: (c

text: b

text: c

text: (d

text: b

text: d

text: k

text: g

text: k

text: g

text: g

text: g

text: g

text: and

text: to

text: the

text: embedded

text: loops

text: γ

text: i

text: (γ

text: b

text: γ

text: (γ

text: b

text: γ

text: m

text: m

text: We

text: call

text: this

text: manifold

text: Y

text: +(G).

text: Similarly,

text: we

text: apply

text: symplectic

text: sum

text: of

text: Y

text: +(G)

text: along

text: the

text: genus

text: two

text: surface

text: Σ

text: descending

text: from

text: Σ

text: ×pt

text: with

text: Z

text: and

text: denote

text: the

text: resulting

text: symplectic

text: manifold

text: by

text: X+(G).

text: Exactly

text: the

text: same

text: argument

text: as

text: in

text: Theorem

text: the

text: abelianization

text: of

text: is

text: Q

text: if

text: the

text: presentation

text: π

text: (X(G)

text: AG

text: G

text: x

text: l

text: l

text: k

text: m

text: is

text: dual

text: ﬁnite

text: torsion.

text: Moreover,

text: by

text: Usher’s

text: theorem

text: X+(G)

text: is

text: minimal.

text: ##LTLine##


text: FUNDAMENTAL GROUP 25 Theorem 3.14. For any integer , there exist minimal symplectic 2 ≤ h ≤ 3 -manifolds and with such that 4 X X p, q ≥ 1 p,q p (1) π (X ) = Z × Z , b+(X ) = 1 and c2(X ) = h; (2) π1(Xp,)q = Z ×p Z , qb+(X ) p=,q 2 and c2(X1) =p,hq . 1 p p p 1 p 4. Cyclic fundamental groups via rational blowdown In this section we present a new technique for constructing symplectic -manifolds with cyclic fundamental groups. The following lemma will be 4 a key technical ingredient in our construction, which we call “the summing spheres”. Proposition 4.1. Let be a symplectic -manifold, is a symplectic torus X 4 T in with self-intersection zero, and . Assume that the spheres , X π (X) = 1 S 1 1 , ... , ( ) are symplectic submanifolds of with self-intersections S S p > 0 X 2 p , , , (where ), each has a single transversal intersection n n · · · n n ≤ 0 S 1 2 p i i with at distinct points, and all intersections are positive. Then, given T the above data, there exists a symplectic 4-manifold Y with c2(Y ) = c2(X), Z 1 1 , such that contains a symplectic sphere of χ (Y ) = χ (X) π (Y ) = Y h h 1 p self-intersection . n + · · · + n 1 p Proof. The manifold will be constructed from by applying Gompf’s Y X symplectic sum operation along T . Consider T2 × S2 equipped with a prod- uct symplectic form, and let denote connected braided symplectic torus T p representing the homology class p[T2 × {pt}] in T2 × S2. We form the sym- plectic sum of X and T2 × S2 along the tori T and T , using a gluing map p that sends the nullhomotopic circles a and a of T to the circles b = bp, 1 2 1 of , in the same order. By Seifert-Van Kampen theorem, the fun- b = c T 2 p damental group of the resulting symplectic manifold can be seen to be Y generated by and . The following relations hold in : and b c π (Y ) c = 1 1 bp = 1. Thus, π (Y ) is isomorphic to Z . The Euler characteristic can be 1 p computed as e(Y ) = e(X) + e(T2 × S2) = e(X), and the Novikov additivity gives σ(Y ) = σ(X) + σ(T2 × S2) = σ(X). When performing the symplectic sum, we have a freedom to identify meridional circles of torus on p α T S i i with p disjoint meridional circles of T on the sphere pt × S2 (See Figure 4). p Thus, we obtain a closed sphere with self-intersection . Finally, S n +· · ·+n 1 p using the Corollary 1.2 ([24], page 17), we conclude that is a symplectic S (cid:3) submanifold of . Y Remark 4.2. If we do symplectic sum the other way, e.g. if we sum up T S one p multiple of the ﬁber in an elliptic ﬁbration with 2 × 2 along ﬁber,

text: FUNDAMENTAL

text: GROUP

text: Theorem

text: For

text: any

text: integer

text: there

text: exist

text: minimal

text: symplectic

text: h

text: -manifolds

text: and

text: with

text: such

text: that

text: X

text: X

text: p,

text: q

text: p,q

text: p

label: list

text: π

text: (X

text: Z

text: Z

text: b+(X

text: and

text: h;

text: Z

text: ×p

text: Z

text: qb+(X

text: p=,q

text: and

text: =p,hq

text: p

text: p

text: p

text: p

label: section

text: Cyclic

text: fundamental

text: groups

text: via

text: rational

text: blowdown

text: In

text: this

text: section

text: we

text: present

text: a

text: new

text: technique

text: for

text: constructing

text: symplectic

text: -manifolds

text: with

text: cyclic

text: fundamental

text: groups.

text: The

text: following

text: lemma

text: will

text: be

text: a

text: key

text: technical

text: ingredient

text: in

text: our

text: construction,

text: which

text: we

text: call

text: “the

text: summing

text: spheres”.

text: Proposition

text: Let

text: be

text: a

text: symplectic

text: -manifold,

text: is

text: a

text: symplectic

text: torus

text: X

text: T

text: in

text: with

text: self-intersection

text: zero,

text: and

text: Assume

text: that

text: the

text: spheres

text: X

text: π

text: (X)

text: S

text: are

text: symplectic

text: submanifolds

text: of

text: with

text: self-intersections

text: S

text: S

text: p

text: X

text: p

text: (where

text: each

text: has

text: a

text: single

text: transversal

text: intersection

text: n

text: n

text: n

text: n

text: S

text: p

text: i

text: i

text: with

text: at

text: distinct

text: points,

text: and

text: all

text: intersections

text: are

text: positive.

text: Then,

text: given

text: T

text: the

text: above

text: data,

text: there

text: exists

text: a

text: symplectic

text: Y

text: with

text: Z

text: such

text: that

text: contains

text: a

text: symplectic

text: sphere

text: of

text: χ

text: (Y

text: χ

text: (X)

text: π

text: (Y

text: Y

text: h

text: h

text: p

text: self-intersection

text: n

text: n

text: p

text: Proof.

text: The

text: manifold

text: will

text: be

text: constructed

text: from

text: by

text: applying

text: Gompf’s

text: Y

text: X

text: symplectic

text: sum

text: operation

text: along

text: T

text: Consider

text: equipped

text: with

text: a

text: prod-

text: uct

text: symplectic

text: form,

text: and

text: let

text: denote

text: connected

text: braided

text: symplectic

text: torus

text: T

text: p

text: representing

text: the

text: homology

text: class

text: {pt}]

text: in

text: We

text: form

text: the

text: sym-

text: plectic

text: sum

text: of

text: X

text: and

text: along

text: the

text: tori

text: T

text: and

text: T

text: using

text: a

text: gluing

text: map

text: p

text: that

text: sends

text: the

text: nullhomotopic

text: circles

text: a

text: and

text: a

text: of

text: T

text: to

text: the

text: circles

text: b

text: bp,

text: of

text: in

text: the

text: same

text: order.

text: By

text: Seifert-Van

text: Kampen

text: theorem,

text: the

text: fun-

text: b

text: c

text: T

text: p

text: damental

text: group

text: of

text: the

text: resulting

text: symplectic

text: manifold

text: can

text: be

text: seen

text: to

text: be

text: Y

text: generated

text: by

text: and

text: The

text: following

text: relations

text: hold

text: in

text: and

text: b

text: c

text: π

text: (Y

text: c

text: bp

text: Thus,

text: π

text: (Y

text: is

text: isomorphic

text: to

text: Z

text: The

text: Euler

text: characteristic

text: can

text: be

text: p

text: computed

text: as

text: e(Y

text: e(X)

text: e(X),

text: and

text: the

text: Novikov

text: additivity

text: gives

text: σ(Y

text: σ(X)

text: σ(X).

text: When

text: performing

text: the

text: symplectic

text: sum,

text: we

text: have

text: a

text: freedom

text: to

text: identify

text: meridional

text: circles

text: of

text: torus

text: on

text: p

text: α

text: T

text: S

text: i

text: i

text: with

text: p

text: disjoint

text: meridional

text: circles

text: of

text: T

text: on

text: the

text: sphere

text: pt

text: (See

text: Figure

text: p

text: Thus,

text: we

text: obtain

text: a

text: closed

text: sphere

text: with

text: self-intersection

text: Finally,

text: S

text: n

text: ·+n

text: p

text: using

text: the

text: Corollary

text: page

text: we

text: conclude

text: that

text: is

text: a

text: symplectic

text: S

text: submanifold

text: of

text: Y

text: Remark

text: If

text: we

text: do

text: symplectic

text: sum

text: the

text: other

text: way,

text: e.g.

text: if

text: we

text: sum

text: up

text: T

text: S

text: one

text: p

text: multiple

text: of

text: the

text: ﬁber

text: in

text: an

text: elliptic

text: ﬁbration

text: with

text: along

text: ﬁber,


text: 28 A. AKHMEDOV AND W. ZHANG C 1 l1 l2 P Q R l3 Figure 5. Conﬁguration of curves in CP2 in the other ﬁbers. See Figure 5. Then we blow up consecutively twice at and , once at the other ﬁve points. We ﬁnally achieve an elliptic P Q E(1) with ﬁber and seven . The homology classes for the components of I I I 5 1 5 are , , , and H − E − E − E H − E − E − E E − E H − E − E − E 1 6 7 2 8 9 2 5 1 2 3 . We thus have seven disjoint sections . E − E E , · · · , E 1 4 3 9 2. We choose a cubic such that it’s zero set has a unique double p C 0 0 point . We also choose another cubic such that it’s zero set is smooth P p C 1 1 and tangent to at the point with multiplicity ﬁve (i.e., this means a - C P 4 0 fold tangency to one branch and transverse to the other branch). The other four intersections are in a general position. Notice that such a conﬁguration exists since there is a cubic curve passing through any given points. Then 6 we choose the points such that four of them are on one branch near , one 6 P point is on the other branch, and the remaining one point is outside of . C 0 If needed, we perturb the obtained curve such that the other intersections 4 with are transversal. Now we can ﬁnd a family of cubics by letting the C 0 ﬁrst of the points vary in family and approach to the point . The 5 6 P limiting curve is . C 1 The family t p + t p for (t , t ) ∈ CP1 gives the pencil structure. We 0 0 1 1 0 1 could choose such that this family tangent to at point (with order p C P 1 1 ﬁve) when , and there are at most ﬁshtail singularities in the other t (cid:54)= 0 0 ﬁbers. See Figure 6. We do ﬁve blow-ups consecutively at and once at each P ##LTFigure##

text: A.

text: AKHMEDOV

text: AND

text: W.

text: ZHANG

text: C

text: P

text: Q

text: R

text: Figure

text: Conﬁguration

text: of

text: curves

text: in

text: in

text: the

text: other

text: ﬁbers.

text: See

text: Figure

text: Then

text: we

text: blow

text: up

text: consecutively

text: twice

text: at

text: and

text: once

text: at

text: the

text: other

text: ﬁve

text: points.

text: We

text: ﬁnally

text: achieve

text: an

text: elliptic

text: P

text: Q

text: with

text: ﬁber

text: and

text: seven

text: The

text: homology

text: classes

text: for

text: the

text: components

text: of

text: I

text: I

text: I

text: are

text: and

text: H

text: E

text: E

text: E

text: H

text: E

text: E

text: E

text: E

text: E

text: H

text: E

text: E

text: E

text: We

text: thus

text: have

text: seven

text: disjoint

text: sections

text: E

text: E

text: E

text: E

text: We

text: choose

text: a

text: cubic

text: such

text: that

text: it’s

text: zero

text: set

text: has

text: a

text: unique

text: double

text: p

text: C

text: point

text: We

text: also

text: choose

text: another

text: cubic

text: such

text: that

text: it’s

text: zero

text: set

text: is

text: smooth

text: P

text: p

text: C

text: and

text: tangent

text: to

text: at

text: the

text: point

text: with

text: multiplicity

text: ﬁve

text: (i.e.,

text: this

text: means

text: a

text: C

text: P

text: fold

text: tangency

text: to

text: one

text: branch

text: and

text: transverse

text: to

text: the

text: other

text: branch).

text: The

text: other

text: four

text: intersections

text: are

text: in

text: a

text: general

text: position.

text: Notice

text: that

text: such

text: a

text: conﬁguration

text: exists

text: since

text: there

text: is

text: a

text: cubic

text: curve

text: passing

text: through

text: any

text: given

text: points.

text: Then

text: we

text: choose

text: the

text: points

text: such

text: that

text: four

text: of

text: them

text: are

text: on

text: one

text: branch

text: near

text: one

text: P

text: point

text: is

text: on

text: the

text: other

text: branch,

text: and

text: the

text: remaining

text: one

text: point

text: is

text: outside

text: of

text: C

text: If

text: needed,

text: we

text: perturb

text: the

text: obtained

text: curve

text: such

text: that

text: the

text: other

text: intersections

text: with

text: are

text: transversal.

text: Now

text: we

text: can

text: ﬁnd

text: a

text: family

text: of

text: cubics

text: by

text: letting

text: the

text: C

text: ﬁrst

text: of

text: the

text: points

text: vary

text: in

text: family

text: and

text: approach

text: to

text: the

text: point

text: The

text: P

text: limiting

text: curve

text: is

text: C

text: The

text: family

text: t

text: p

text: t

text: p

text: for

text: (t

text: t

text: gives

text: the

text: pencil

text: structure.

text: We

text: could

text: choose

text: such

text: that

text: this

text: family

text: tangent

text: to

text: at

text: point

text: (with

text: order

text: p

text: C

text: P

text: ﬁve)

text: when

text: and

text: there

text: are

text: at

text: most

text: ﬁshtail

text: singularities

text: in

text: the

text: other

text: t

text: ﬁbers.

text: See

text: Figure

text: We

text: do

text: ﬁve

text: blow-ups

text: consecutively

text: at

text: and

text: once

text: at

text: each

text: P

text: ##LTFigure##


text: 34 A. AKHMEDOV AND W. ZHANG ﬁbration on , and thus we choose vanishing cycles for the circles on the E(2) ﬁber in corresponding to , . We will still call these curves E(2) a 1 ≤ i ≤ 4 i . Since we have singular ﬁbers in the generic elliptic ﬁbration of a 12 E(2) i corresponding to the class . Take diﬀerent singular ﬁbers out of these , a 4 12 which will give us disjoint vanishing cycles intersecting the chosen ﬁber at 4 respectively. Furthemore, using the given odering of these vanishing cycles a i and their corresponding vanishing discs, we construct four matching spheres , , , and in the symplectic -manifold . S(a , a ) S(a , a ) S(a , a ) S(a , a ) 4 X 1 2 2 3 3 4 4 1 Each of these spheres contains one and only one intersection points with −4 spheres as mentioned above. Hence the meridians of these four spheres −4 would be trivial in . Since Z , the fundamental group of the π (X) π (X) = 1 1 2 manifold after rationally blowing down the four spheres is still Z . −4 2 Acknowledgments A. Akhmedov was partially supported by NSF grants FRG-1065955 and DMS-1005741. W. Zhang was partially supported by AMS-Simons travel grant. We would like to thank I. Dolgachev for helpful discussions. We are grateful to Tian-Jun Li for suggestions and for his kind encouragement. The authors are also thankful to Rafael Torres for his remarks and suggestions. Finally, we would like to thank the anonymous referees who gave us many constructive suggestions and valuable comments on the previous versions of this paper, which led to the improvement of the presentation of the material in the paper. References [1] A. Akhmedov, Construction of symplectic cohomology S2 ×S2, Gökova Geometry and Topology Proceedings, 14 (2007), 36–48. [2] A. Akhmedov, Small exotic 4-manifolds, Algebr. Geom. Topol. 8 (2008), 1781–1794. [3] A. Akhmedov, R.I. Baykur, B.D. Park, Constructing inﬁnitely many smooth struc- tures on small 4-manifolds, J. Topol. 1 (2008), no. 2, 409–428. [4] A. Akhmedov, B. D. Park, Exotic 4-manifolds with small Euler characteristic, Inven. Math, 173 (2008), no.1, 209–223. [5] A. Akhmedov, B. D. Park, Exotic Smooth Structures on Small 4-Manifolds with Odd Signatures, Invent. Math. 181 (2010), no. 3, 577–603. [6] A. Akhmedov, B. D Park, Geography of simply connected spin symplectic 4-manifolds, Math. Res. Lett. 17 (2010), 483–492. [7] A. Akhmedov, S. Baldridge, R. İ. Baykur, P. Kirk and B. D. Park, Simply connected − minimal symplectic 4-manifolds with signature less than 1, J. Eur. Math. Soc. 1 (2010), 133–161. [8] A. Akhmedov and B. Ozbagci: Exotic Stein ﬁllings with arbitrary fundamental group, preprint, arXiv:1212.1743. [9] J. Amorós, F. Bogomolov, L. Katzarkov, T. Pantev, Symplectic Lefschetz ﬁbrations with arbitrary fundamental groups, J. Diﬀerential Geom. 54 (2000), no. 3, 489–545.

text: A.

text: AKHMEDOV

text: AND

text: W.

text: ZHANG

text: ﬁbration

text: on

text: and

text: thus

text: we

text: choose

text: vanishing

text: cycles

text: for

text: the

text: circles

text: on

text: the

text: ﬁber

text: in

text: corresponding

text: to

text: We

text: will

text: still

text: call

text: these

text: curves

text: a

text: i

text: i

text: Since

text: we

text: have

text: singular

text: ﬁbers

text: in

text: the

text: generic

text: elliptic

text: ﬁbration

text: of

text: a

text: i

text: corresponding

text: to

text: the

text: class

text: Take

text: diﬀerent

text: singular

text: ﬁbers

text: out

text: of

text: these

text: a

text: which

text: will

text: give

text: us

text: disjoint

text: vanishing

text: cycles

text: intersecting

text: the

text: chosen

text: ﬁber

text: at

text: respectively.

text: Furthemore,

text: using

text: the

text: given

text: odering

text: of

text: these

text: vanishing

text: cycles

text: a

text: i

text: and

text: their

text: corresponding

text: vanishing

text: discs,

text: we

text: construct

text: four

text: matching

text: spheres

text: and

text: in

text: the

text: symplectic

text: -manifold

text: S(a

text: a

text: S(a

text: a

text: S(a

text: a

text: S(a

text: a

text: X

text: Each

text: of

text: these

text: spheres

text: contains

text: one

text: and

text: only

text: one

text: intersection

text: points

text: with

text: −4

text: spheres

text: as

text: mentioned

text: above.

text: Hence

text: the

text: meridians

text: of

text: these

text: four

text: spheres

text: −4

text: would

text: be

text: trivial

text: in

text: Since

text: Z

text: the

text: fundamental

text: group

text: of

text: the

text: π

text: (X)

text: π

text: (X)

text: manifold

text: after

text: rationally

text: blowing

text: down

text: the

text: four

text: spheres

text: is

text: still

text: Z

text: −4

text: Acknowledgments

text: A.

text: Akhmedov

text: was

text: partially

text: supported

text: by

text: NSF

text: grants

text: and

text: W.

text: Zhang

text: was

text: partially

text: supported

text: by

text: AMS-Simons

text: travel

text: grant.

text: We

text: would

text: like

text: to

text: thank

text: I.

text: Dolgachev

text: for

text: helpful

text: discussions.

text: We

text: are

text: grateful

text: to

text: Tian-Jun

text: Li

text: for

text: suggestions

text: and

text: for

text: his

text: kind

text: encouragement.

text: The

text: authors

text: are

text: also

text: thankful

text: to

text: Rafael

text: Torres

text: for

text: his

text: remarks

text: and

text: suggestions.

text: Finally,

text: we

text: would

text: like

text: to

text: thank

text: the

text: anonymous

text: referees

text: who

text: gave

text: us

text: many

text: constructive

text: suggestions

text: and

text: valuable

text: comments

text: on

text: the

text: previous

text: versions

text: of

text: this

text: paper,

text: which

text: led

text: to

text: the

text: improvement

text: of

text: the

text: presentation

text: of

text: the

text: material

text: in

text: the

text: paper.

text: References

text: A.

text: Akhmedov,

text: Construction

text: of

text: symplectic

text: cohomology

text: Gökova

text: Geometry

text: and

text: Topology

text: Proceedings,

text: A.

text: Akhmedov,

text: Small

text: exotic

text: Algebr.

text: Geom.

text: Topol.

text: A.

text: Akhmedov,

text: R.I.

text: Baykur,

text: B.D.

text: Park,

text: Constructing

text: inﬁnitely

text: many

text: smooth

text: struc-

text: tures

text: on

text: small

text: J.

text: Topol.

text: no.

text: A.

text: Akhmedov,

text: B.

text: D.

text: Park,

text: Exotic

text: with

text: small

text: Euler

text: characteristic,

text: Inven.

text: Math,

text: A.

text: Akhmedov,

text: B.

text: D.

text: Park,

text: Exotic

text: Smooth

text: Structures

text: on

text: Small

text: with

text: Odd

text: Signatures,

text: Invent.

text: Math.

text: no.

text: A.

text: Akhmedov,

text: B.

text: D

text: Park,

text: Geography

text: of

text: simply

text: connected

text: spin

text: symplectic

text: Math.

text: Res.

text: Lett.

text: A.

text: Akhmedov,

text: S.

text: Baldridge,

text: R.

text: İ.

text: Baykur,

text: P.

text: Kirk

text: and

text: B.

text: D.

text: Park,

text: Simply

text: connected

text: minimal

text: symplectic

text: with

text: signature

text: less

text: than

text: J.

text: Eur.

text: Math.

text: Soc.

text: A.

text: Akhmedov

text: and

text: B.

text: Ozbagci:

text: Exotic

text: Stein

text: ﬁllings

text: with

text: arbitrary

text: fundamental

text: group,

text: preprint,

text: J.

text: Amorós,

text: F.

text: Bogomolov,

text: L.

text: Katzarkov,

text: T.

text: Pantev,

text: Symplectic

text: Lefschetz

text: ﬁbrations

text: with

text: arbitrary

text: fundamental

text: groups,

text: J.

text: Diﬀerential

text: Geom.

text: no.