text: Lemma 10: For the ﬁrst term I in (74), − (cid:48) ≤ 6 2(1 γ)g (γ) I + I + I + I + I . (69) 1 5 6 3 4 ≤ I 0. (75) 6 We now upper bound the ﬁrst two terms in r.h.s of (69). Lemma 9: For the terms I and I , By data processing inequality in Lemma 4, 1 5 ≤ I + I I , (70) | − − ∗ 1 5 7 VT J(X(cid:112) M )V VT (K 1 + B )V L L,γ L L L L L (cid:23) − − ∗ whe(cid:88)re (cid:110) =VVLTT JJ((XL1,γ−)VγXL +V√LTγ(XKG)1V+ B−LV)VT (LK−1 + B∗ )V L L L L L L − − ∗ − − =VT ((1 γ)K + γ(K 1 + B ) 1) 1V L (cid:16) (cid:17)(cid:111) L− − ∗ L L I7 = tr 2UiUTi ∆iUiUTi VLT (K 1 + BL)VL − − − ∗ − − ∗ i=1· − − ∗ − − | − =VLT K 1(K 1 + (1 γ)BL) 1(K 1 + BL)VL (Σ 1 B ) Σ 1 cov(Y X, M , Q)Σ 1 . − − ∗ i i i i,γ i i VT (K 1 + B )V (71) L L L =0, (76) It follows from Proposition 3 in Theorem 2 that ∗ where the last step comes from B V = 0 in Proposition (cid:31) − L L WT ∆ W 0, i = 1, . . . , L 1. (72) 4 in Theorem 2. i i+1 i On the other hand, by Proposition 3 in Theorem 2, (cid:22) VT ∆(cid:110)V 0, we see th(cid:16)at (cid:17)(cid:111) On the other hand, L L L (cid:110) (cid:111) | − − ∗ | − − tr V VT ∆ V VT J(X M ) (K 1 + B ) J(X M , . . . , M ) C 1 L L L L L L,γ L L i+1,γ i+1 L i+1  ≤ · ((cid:22)a)(1 − γ)J(X(cid:88)|Mi+(cid:0)1, . . . , ML) − (1 − γ)C−i+11 (cid:1) tr VLT ∆LVL 0 = 0. ((cid:22)b)(1 − γ) L(cid:0) Σ−j 1 − Σ−j(cid:1)1 cov(Yj|X, Mj, Q)Σ−j 1 Lemma 11: For the second term I7 and the third term I3 in (74), j=i+1 (cid:88)− − − − (1 γ) K 1 + C 1 ≤ i+1 I + I 0. (77) (=c) L (Σ−1 − B∗) − Σ−1 cov(Y |X, M , Q)Σ−1, 7 3 j j j j,γ j j j=i+1 By the deﬁnition of ∆ : (73) i (cid:44) µ − − ∗ − − ∆ i (Σ 1 B ) 1 Ψ , where (a) follows from the deﬁnition of random vector i 2 i i i { } Xi+1,γ and Fisher information inequality in Lemma 5, (b) we can write I7 + I3 in the following form: can proved by using the argument in [9, Section 6.2] (for completeness, we rewrite the proof in [9] in Appendix C), (cid:88) (cid:110) { } and (c) is due to the deﬁnition of random vector Yj,γ . I7 + I3 Finally, we obtain the bound (70) by substituting (72) − L (cid:16) (cid:17)(cid:111) (73) into I5 and (73) into I1 then simplifying it using the = tr 2UiUTi ΨiUiUTi relationship ( Proposition 2 in Theorem 2 ): i=1 · − − ∗ − − | − (Σ 1 B ) Σ 1 cov(Y X, M , Q)Σ 1 i i i i,γ i i (78) U UT ∆ U UT i+1 i+1 i+1 i i = UiUTi ∆i+1UiUTi + WiWiT ∆i+1WiWiT , Considering that i = 1, . . . , L. | cov(Y X, M , Q) i,γ i Substituting the upper bound (70) into (69) gives (a) − | − ∗ =(1 γ) cov(Y X, M , Q) + γ(Σ Σ B Σ ) − (cid:48) ≤ (cid:22) − i| i − i ∗ i i i 2(1 γ)g (γ) I6 + I7 + I3 + I4. (74) (1 −γ) cov(∗Yi X) + γ(Σi ΣiBi Σi) =Σ γΣ B Σ , (79) i i i i We now upper bound each term separately. 10 ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: For

text: the

text: ﬁrst

text: term

text: I

text: in

text: γ)g

text: (γ)

text: I

text: I

text: I

text: I

text: I

text: I

text: We

text: now

text: upper

text: bound

text: the

text: ﬁrst

text: two

text: terms

text: in

text: r.h.s

text: of

text: Lemma

text: For

text: the

text: terms

text: I

text: and

text: I

text: By

text: data

text: processing

text: inequality

text: in

text: Lemma

text: I

text: I

text: I

text: VT

text: M

text: )V

text: VT

text: (K

text: B

text: )V

text: L

text: L,γ

text: L

text: L

text: L

text: L

text: L

text: =VVLTT

text: B−LV)VT

text: B∗

text: )V

text: L

text: L

text: L

text: L

text: L

text: L

text: =VT

text: γ)K

text: γ(K

text: B

text: L

text: L−

text: L

text: L

text: tr

text: ∆iUiUTi

text: VLT

text: (K

text: BL)VL

text: =VLT

text: K

text: γ)BL)

text: BL)VL

text: (Σ

text: B

text: Σ

text: cov(Y

text: X,

text: M

text: Q)Σ

text: i

text: i

text: i

text: i,γ

text: i

text: i

text: VT

text: (K

text: B

text: )V

text: L

text: L

text: L

text: It

text: follows

text: from

text: Proposition

text: in

text: Theorem

text: that

text: where

text: the

text: last

text: step

text: comes

text: from

text: B

text: V

text: in

text: Proposition

text: L

text: L

text: WT

text: W

text: i

text: L

text: in

text: Theorem

text: i

text: i

text: On

text: the

text: other

text: hand,

text: by

text: Proposition

text: in

text: Theorem

text: VT

text: we

text: see

text: On

text: the

text: other

text: hand,

text: L

text: L

text: L

text: tr

text: V

text: VT

text: V

text: VT

text: J(X

text: M

text: (K

text: B

text: J(X

text: M

text: M

text: C

text: L

text: L

text: L

text: L

text: L

text: L,γ

text: L

text: L

text: L

text: ML)

text: tr

text: VLT

text: ∆LVL

text: γ)

text: Σ−j

text: cov(Yj|X,

text: Mj,

text: Q)Σ−j

text: Lemma

text: For

text: the

text: second

text: term

text: and

text: the

text: third

text: term

text: in

text: γ)

text: K

text: C

text: I

text: I

text: (=c)

text: L

text: (Σ−1

text: B∗)

text: Σ−1

text: cov(Y

text: |X,

text: M

text: j

text: j

text: j

text: j,γ

text: j

text: j

text: By

text: the

text: deﬁnition

text: of

text: i

text: µ

text: i

text: (Σ

text: B

text: Ψ

text: where

text: (a)

text: follows

text: from

text: the

text: deﬁnition

text: of

text: random

text: vector

text: i

text: i

text: i

text: i

text: and

text: Fisher

text: information

text: inequality

text: in

text: Lemma

text: (b)

text: we

text: can

text: write

text: in

text: the

text: following

text: form:

text: can

text: proved

text: by

text: using

text: the

text: argument

text: in

text: Section

text: (for

text: completeness,

text: we

text: rewrite

text: the

text: proof

text: in

text: in

text: Appendix

text: C),

text: and

text: (c)

text: is

text: due

text: to

text: the

text: deﬁnition

text: of

text: random

text: vector

text: Yj,γ

text: Finally,

text: we

text: obtain

text: the

text: bound

text: by

text: substituting

text: L

text: into

text: and

text: into

text: then

text: simplifying

text: it

text: using

text: the

text: tr

text: ΨiUiUTi

text: relationship

text: Proposition

text: in

text: Theorem

text: (Σ

text: B

text: Σ

text: cov(Y

text: X,

text: M

text: Q)Σ

text: i

text: i

text: i

text: i,γ

text: i

text: i

text: U

text: UT

text: U

text: UT

text: i

text: i

text: UiUTi

text: WiWiT

text: Considering

text: that

text: i

text: L.

text: cov(Y

text: X,

text: M

text: Q)

text: i,γ

text: i

text: Substituting

text: the

text: upper

text: bound

text: into

text: gives

text: (a)

text: γ)

text: cov(Y

text: X,

text: M

text: Q)

text: γ(Σ

text: Σ

text: B

text: Σ

text: i|

text: i

text: i

text: i

text: i

text: i

text: γ)g

text: (γ)

text: −γ)

text: cov(∗Yi

text: X)

text: γ(Σi

text: ΣiBi

text: Σi)

text: =Σ

text: γΣ

text: B

text: Σ

text: i

text: i

text: i

text: i

text: We

text: now

text: upper

text: bound

text: each

text: term

text: separately.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 TWISTED SQFT Lemma 2.17. The suture elements s(Γ) are respected: for any sutures Γ on (Σ, V ), and decorated −→ (cid:48) (cid:48) morphism (φ, Γ ) : (Σ, V ) (Σ , V ), c D ∪ s(Γ) = s(Γ Γ ). φ,Γc c (cid:48) Proof. Taking a small disc on (Σ, V ) intersecting Γ in an arc, and its image under φ on Σ , we (cid:48) (cid:48) ∅ (cid:48) obtain vacuum inclusions into (Σ, V ) and (Σ , V ), which send sutures Γ on the vacuum to Γ and Γ respectively, so that the following diagram commutes. ∅ ∅ (Σ , V ) (cid:48) ι ι (φ, Γc) (cid:48) (cid:48) (Σ, V ) (Σ , V ) (cid:48) D ∅ D D ∅ D We then have, by deﬁnition s(Γ ) = (cid:48)s(Γ ) = s(Γ ) = s(Γ). ι φ,Γc ι φ,Γc We can now complete the proof of nonexistence of sanguine twisted SQFT along similar lines as the optimistic case. Proof of proposition 2.12. As in the proof of proposition 2.10 Let (Σ, V ) be a quadrangulated disc with V Z 6 vertices. We have (Σ, V ; 0) 2-dimensional over with ordered basis (s(Γ− ), s(Γ −)). + + Considering again the rotation morphism R and the corresponding module homomorphism in (cid:55)→ grading 0, since R sends Γ− Γ − and R3 sends sutures to themselves, we again obtain s(Γ− ) + + + + s(Γ −) + s(Γ±) = 0. + −→ (cid:3) (cid:3) D And considering (Φ, Γ ) : (Σ, V ) (Σ , V ) again we ﬁnd sends c Φ,Γc (cid:55)→ (cid:55)→ − − (cid:55)→ s(Γ− ) s(Γ ), s(Γ −) 0, s(Γ− ) s(Γ −) s(Γ ). + + + + + + ∈ {± } ⊂ Z Adding these gives 2s(Γ ) = 0, a contradiction as s(Γ ) c(Γ ) = 1 . + + + This completes the proof of proposition 1.2. Twisted SQFT is therefore consigned to eternal ambiguity; but only up to units, which is not a particularly costly price to pay. 2.4 Deﬁnition of twisted SQFT Having learned our lesson from the previous section, we cannot obtain a bona ﬁde functor from DOS, the decorated occupied surface category, to a graded module category. We consider graded module homomorphisms up to units, and suture elements up to units. A twisted SQFT is thus a “ functor up to units”. When we speak of an element of an R-module M “up to units”, we really mean the orbit of an × element of M under the multiplicative action of the group of units R . We require each c(Γ) to be Z × V such an orbit, i.e. a [H (Σ)] -orbit in (Σ, V ). 1 The following lemma shows that, even when module homomorphisms are ambiguous up to units, they never mix up such orbits. (cid:48) (cid:48) −→ (cid:48) Lemma 2.18. Let M, M be graded modules over rings R, R , and let f : M M be a graded ∈ × module homomorphism over a ring homomorphism f¯. If m , m M lie in the same R -orbit, then (cid:48)× 1 ×2 (cid:48)× (cid:48) f(m ), f(m ) lie in the same R -orbit. That is, f sends an R -orbit of M into an R -orbit of M . 1 2 (cid:48) ∈ × (cid:48) Proof. If m = rm for some r R , then f(m ) = f(rm) = f¯(r) f(m), where f¯(r) is a unit, as any ring homomorphism preserves units. 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: TWISTED

text: SQFT

text: Lemma

text: The

text: suture

text: elements

text: s(Γ)

text: are

text: respected:

text: for

text: any

text: sutures

text: Γ

text: on

text: (Σ,

text: V

text: and

text: decorated

text: morphism

text: (φ,

text: Γ

text: (Σ,

text: V

text: (Σ

text: V

text: c

text: D

text: s(Γ)

text: s(Γ

text: Γ

text: φ,Γc

text: c

text: Proof.

text: Taking

text: a

text: small

text: disc

text: on

text: (Σ,

text: V

text: intersecting

text: Γ

text: in

text: an

text: arc,

text: and

text: its

text: image

text: under

text: φ

text: on

text: Σ

text: we

text: obtain

text: vacuum

text: inclusions

text: into

text: (Σ,

text: V

text: and

text: (Σ

text: V

text: which

text: send

text: sutures

text: Γ

text: on

text: the

text: vacuum

text: to

text: Γ

text: and

text: Γ

text: respectively,

text: so

text: that

text: the

text: following

text: diagram

text: commutes.

text: (Σ

text: V

text: ι

text: ι

text: (φ,

text: Γc)

text: (Σ,

text: V

text: (Σ

text: V

text: D

text: D

text: D

text: D

text: We

text: then

text: have,

text: by

text: deﬁnition

text: s(Γ

text: s(Γ

text: s(Γ).

text: ι

text: φ,Γc

text: ι

text: φ,Γc

text: We

text: can

text: now

text: complete

text: the

text: proof

text: of

text: nonexistence

text: of

text: sanguine

text: twisted

text: SQFT

text: along

text: similar

text: lines

text: as

text: the

text: optimistic

text: case.

text: Proof

text: of

text: proposition

text: As

text: in

text: the

text: proof

text: of

text: proposition

text: Let

text: (Σ,

text: V

text: be

text: a

text: quadrangulated

text: disc

text: with

text: V

text: Z

text: vertices.

text: We

text: have

text: (Σ,

text: V

text: over

text: with

text: ordered

text: basis

text: (s(Γ−

text: s(Γ

text: Considering

text: again

text: the

text: rotation

text: morphism

text: R

text: and

text: the

text: corresponding

text: module

text: homomorphism

text: in

text: grading

text: since

text: R

text: sends

text: Γ−

text: Γ

text: and

text: sends

text: sutures

text: to

text: themselves,

text: we

text: again

text: obtain

text: s(Γ−

text: s(Γ

text: s(Γ±)

text: D

text: And

text: considering

text: (Φ,

text: Γ

text: (Σ,

text: V

text: (Σ

text: V

text: again

text: we

text: ﬁnd

text: sends

text: c

text: Φ,Γc

text: s(Γ−

text: s(Γ

text: s(Γ

text: s(Γ−

text: s(Γ

text: s(Γ

text: Z

text: Adding

text: these

text: gives

text: a

text: contradiction

text: as

text: s(Γ

text: c(Γ

text: This

text: completes

text: the

text: proof

text: of

text: proposition

text: Twisted

text: SQFT

text: is

text: therefore

text: consigned

text: to

text: eternal

text: ambiguity;

text: but

text: only

text: up

text: to

text: units,

text: which

text: is

text: not

text: a

text: particularly

text: costly

text: price

text: to

text: pay.

text: Deﬁnition

text: of

text: twisted

text: SQFT

text: Having

text: learned

text: our

text: lesson

text: from

text: the

text: previous

text: section,

text: we

text: cannot

text: obtain

text: a

text: bona

text: ﬁde

text: functor

text: from

text: DOS,

text: the

text: decorated

text: occupied

text: surface

text: category,

text: to

text: a

text: graded

text: module

text: category.

text: We

text: consider

text: graded

text: module

text: homomorphisms

text: up

text: to

text: units,

text: and

text: suture

text: elements

text: up

text: to

text: units.

text: A

text: twisted

text: SQFT

text: is

text: thus

text: a

text: functor

text: up

text: to

text: units”.

text: When

text: we

text: speak

text: of

text: an

text: element

text: of

text: an

text: R-module

text: M

text: “up

text: to

text: units”,

text: we

text: really

text: mean

text: the

text: orbit

text: of

text: an

text: element

text: of

text: M

text: under

text: the

text: multiplicative

text: action

text: of

text: the

text: group

text: of

text: units

text: R

text: We

text: require

text: each

text: c(Γ)

text: to

text: be

text: Z

text: V

text: such

text: an

text: orbit,

text: i.e.

text: a

text: [H

text: (Σ)]

text: -orbit

text: in

text: (Σ,

text: V

text: The

text: following

text: lemma

text: shows

text: that,

text: even

text: when

text: module

text: homomorphisms

text: are

text: ambiguous

text: up

text: to

text: units,

text: they

text: never

text: mix

text: up

text: such

text: orbits.

text: Lemma

text: Let

text: M,

text: M

text: be

text: graded

text: modules

text: over

text: rings

text: R,

text: R

text: and

text: let

text: f

text: M

text: M

text: be

text: a

text: graded

text: module

text: homomorphism

text: over

text: a

text: ring

text: homomorphism

text: f¯.

text: If

text: m

text: m

text: M

text: lie

text: in

text: the

text: same

text: R

text: -orbit,

text: then

text: ×2

text: f(m

text: f(m

text: lie

text: in

text: the

text: same

text: R

text: -orbit.

text: That

text: is,

text: f

text: sends

text: an

text: R

text: -orbit

text: of

text: M

text: into

text: an

text: R

text: -orbit

text: of

text: M

text: Proof.

text: If

text: m

text: rm

text: for

text: some

text: r

text: R

text: then

text: f(m

text: f(rm)

text: f¯(r)

text: f(m),

text: where

text: f¯(r)

text: is

text: a

text: unit,

text: as

text: any

text: ring

text: homomorphism

text: preserves

text: units.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 TWISTED SQFT multiplying by a unit in each summand in this way. These form an equivalence class of graded module D homomorphisms and strictly speaking is this collection of homomorphisms. φ,Γc D “Functor up to units” and categorical considerations. The ﬁrst property of expresses precisely Z “functoriality up to units”. In the untwisted case, there are no units other than 1, so “up to units” 2 D becomes redundant, and SQFT is a bona ﬁde functor. It would be more satisfactory if could be expressed cleanly as a bona ﬁde functor between appropriate “quotient” categories; this can be done, but involves greater technical detail, and we leave it to subsequent work. V Naturality of (Σ, V ). Considering decorated morphisms which are homeomorphisms, functoriality V up to units gives (Σ, V ) properties of “naturality up to units” (see [12] for a full discussion of −→ (cid:48) (cid:48) D naturality). In particular, if φ : (Σ, V ) (Σ , V ) is a homeomorphism then ∅ is an isomorphism. φ, D (More precisely, an equivalence class up to units of isomorphisms.) This follows since respects ∅ − ∅ composition and identity up to units, and since (φ, ) has inverse decorated morphism (φ 1, ). D Decorated-isotopy classes. We have speciﬁed, unlike in [17], that the homomorphisms only Z φ,Γc depend on the decorated-isotopy class of (φ, Γ ). In the case the uniqueness of suture elements c 2 made the invariance of homomorphisms under decorated isotopy clear [17, lemma 8.3]. It turns out (cid:78) also in the twisted case that imposing decorated-isotopy invariance is redundant, but this requires further work which we defer to the seqVuel. ∼ Z ⊗ V (cid:3) (cid:3) V Free modules and bases. Writing (Σ, V ) = [H (Σ)] (Σ , V ), expresses (Σ, V ) as a Z V (cid:3) (cid:3) 1 i iZ i free module over [H (Σ)]. Each (Σ , V ) is a free 2-dimensional -module; their tensor product Z 1 Z i i over is a 2I(Σ,V )-dimensional -module (recall I(Σ, V ) is the number of squares); taking the tensor Z Z product with [H (Σ)] gives a free 2I(Σ,V )-dimensional [H (Σ)]-module. Given a quadrangulation Q 1 V Z 1 of (Σ, V ), we naturally obtain a basis of (Σ, V ) over [H (Σ, V )]. Let the I(Σ, V ) squares of Q be (cid:3) (cid:3) V (cid:3) (cid:3) 1 V Z (Σ , V ), and take a basis 0, 1 of each (Σ , V ). Then a basis of (Σ, V ) over [H (Σ)] is given i i i i 1 by ⊗ ⊗ · · · ⊗ | · · · (cid:105) e e e = e e e , 1 2 I(Σ,V ) 1 2 I(Σ,V ) where each ei is 0 or 1. These 2I(Σ,V ) basis elements consist of a representative of e(cid:76)ach cV(Γ), over∼aZll basic sets of sutures Γ. When (Σ, V ) is the vacuum with the null quadrangulation, we have (Σ, V ) = Z and we write a basis over simply as 1 (not 1). V V Bases for graded components. Writing the graded decomposition of (Σ, V ) as (Σ, V ; e), we V e then obtain a basis for each (Σ, V ; e). Let n , n denote the number of 0’s and 1’s respectively among V 0 1Z | · · · (cid:105) e , . . . , e . Then (Σ, V ; e) has basis over [H (Σ)] given by those e e e satisfying 1 − I(Σ,V ) 1 1 2 I(Σ,V ) n n = e. These basis elements are precisely those which are suture elements for basic sutures 1 0 V with Euler class e. It follows that (Σ, V ; e) is nontrivial precisely for e of the same parity as I(Σ, V ) | | ≤ and satisfying e I(Σ, V ). This corresponds to the fact that e(Γ) satisﬁes the same properties, for nontrivial sutures Γ. (cid:48) Sutures up to homeomorphism vs isotopy. Two sets of sutures Γ, Γ on (Σ, V ) may be homeomorphic −→ (cid:48) (cid:48) in the sense that there is a homeomorphism φ : Σ Σ taking Γ to Γ ; but such Γ, Γ are in general not isotopic. For instance, if Γ is a set of sutures on a disc, and we rotate the disc, the resulting sutures (cid:48) Γ will in general be non-isotopic. Suture elements c(Γ) are invariant under isotopy of sutures, but in (cid:48) general not under homeomorphism; so c(Γ), c(Γ ) will in general be diﬀerent. D V −→ Keeping track of suture element orbits. We saw in lemma 2.18 that each : (Σ, V ) V (cid:48) (cid:48) Z × Z (cid:48) × φ,Γc Z −→ (Σ , V ) sends a [H (Σ)] -orbit into a [H (Σ )] -orbit. Note, however, that φ∗ : [H (Σ)] Z (cid:48) 1 1 D 1 [H (Σ )] might not be surjective, and so the image of a c(Γ) under might not be an entire Z 1 (cid:48) × ∪ ⊆ (cid:48) φ,Γc [H (Σ )] -orbit. This is why we write c(Γ Γ ) c(Γ ), rather than equality. Given the ambiguity D1 D c (cid:48) in , however, we may post-multiply by a unit to hit any given element of c(Γ ). φ,Γc φ,Γc D Gradings of decorated morphisms. Finally, we note that the degree of each is given by the φ,Γc amount by which is shifts Euler classes, i.e. the degree of the decorated morphism deg(φ, Γ ). c 2.5 First properties of suture elements We can now develop some basic properties of suture elements. 19 ##LTLine##

text: TWISTED

text: SQFT

text: multiplying

text: by

text: a

text: unit

text: in

text: each

text: summand

text: in

text: this

text: way.

text: These

text: form

text: an

text: equivalence

text: class

text: of

text: graded

text: module

text: D

text: homomorphisms

text: and

text: strictly

text: speaking

text: is

text: this

text: collection

text: of

text: homomorphisms.

text: φ,Γc

text: D

text: “Functor

text: up

text: to

text: units”

text: and

text: categorical

text: considerations.

text: The

text: ﬁrst

text: property

text: of

text: expresses

text: precisely

text: Z

text: “functoriality

text: up

text: to

text: units”.

text: In

text: the

text: untwisted

text: case,

text: there

text: are

text: no

text: units

text: other

text: than

text: so

text: “up

text: to

text: units”

text: D

text: becomes

text: redundant,

text: and

text: SQFT

text: is

text: a

text: bona

text: ﬁde

text: functor.

text: It

text: would

text: be

text: more

text: satisfactory

text: if

text: could

text: be

text: expressed

text: cleanly

text: as

text: a

text: bona

text: ﬁde

text: functor

text: between

text: appropriate

text: “quotient”

text: categories;

text: this

text: can

text: be

text: done,

text: but

text: involves

text: greater

text: technical

text: detail,

text: and

text: we

text: leave

text: it

text: to

text: subsequent

text: work.

text: V

text: Naturality

text: of

text: (Σ,

text: V

text: Considering

text: decorated

text: morphisms

text: which

text: are

text: homeomorphisms,

text: functoriality

text: V

text: up

text: to

text: units

text: gives

text: (Σ,

text: V

text: properties

text: of

text: “naturality

text: up

text: to

text: units”

text: (see

text: for

text: a

text: full

text: discussion

text: of

text: D

text: naturality).

text: In

text: particular,

text: if

text: φ

text: (Σ,

text: V

text: (Σ

text: V

text: is

text: a

text: homeomorphism

text: then

text: is

text: an

text: isomorphism.

text: φ,

text: D

text: (More

text: precisely,

text: an

text: equivalence

text: class

text: up

text: to

text: units

text: of

text: isomorphisms.)

text: This

text: follows

text: since

text: respects

text: composition

text: and

text: identity

text: up

text: to

text: units,

text: and

text: since

text: (φ,

text: has

text: inverse

text: decorated

text: morphism

text: (φ

text: D

text: Decorated-isotopy

text: classes.

text: We

text: have

text: speciﬁed,

text: unlike

text: in

text: that

text: the

text: homomorphisms

text: only

text: Z

text: φ,Γc

text: depend

text: on

text: the

text: decorated-isotopy

text: class

text: of

text: (φ,

text: Γ

text: In

text: the

text: case

text: the

text: uniqueness

text: of

text: suture

text: elements

text: c

text: made

text: the

text: invariance

text: of

text: homomorphisms

text: under

text: decorated

text: isotopy

text: clear

text: lemma

text: It

text: turns

text: out

text: also

text: in

text: the

text: twisted

text: case

text: that

text: imposing

text: decorated-isotopy

text: invariance

text: is

text: redundant,

text: but

text: this

text: requires

text: further

text: work

text: which

text: we

text: defer

text: to

text: the

text: seqVuel.

text: Z

text: V

text: V

text: Free

text: modules

text: and

text: bases.

text: Writing

text: (Σ,

text: V

text: [H

text: (Σ)]

text: (Σ

text: V

text: expresses

text: (Σ,

text: V

text: as

text: a

text: Z

text: V

text: i

text: iZ

text: i

text: free

text: module

text: over

text: [H

text: (Σ)].

text: Each

text: (Σ

text: V

text: is

text: a

text: free

text: -module;

text: their

text: tensor

text: product

text: Z

text: Z

text: i

text: i

text: over

text: is

text: a

text: )-dimensional

text: -module

text: (recall

text: I(Σ,

text: V

text: is

text: the

text: number

text: of

text: squares);

text: taking

text: the

text: tensor

text: Z

text: Z

text: product

text: with

text: [H

text: (Σ)]

text: gives

text: a

text: free

text: )-dimensional

text: [H

text: (Σ)]-module.

text: Given

text: a

text: quadrangulation

text: Q

text: V

text: Z

text: of

text: (Σ,

text: V

text: we

text: naturally

text: obtain

text: a

text: basis

text: of

text: (Σ,

text: V

text: over

text: [H

text: (Σ,

text: V

text: Let

text: the

text: I(Σ,

text: V

text: squares

text: of

text: Q

text: be

text: V

text: V

text: Z

text: (Σ

text: V

text: and

text: take

text: a

text: basis

text: of

text: each

text: (Σ

text: V

text: Then

text: a

text: basis

text: of

text: (Σ,

text: V

text: over

text: [H

text: (Σ)]

text: is

text: given

text: i

text: i

text: i

text: i

text: by

text: e

text: e

text: e

text: e

text: e

text: e

text: I(Σ,V

text: I(Σ,V

text: where

text: each

text: ei

text: is

text: or

text: These

text: basis

text: elements

text: consist

text: of

text: a

text: representative

text: of

text: cV(Γ),

text: over∼aZll

text: basic

text: sets

text: of

text: sutures

text: Γ.

text: When

text: (Σ,

text: V

text: is

text: the

text: vacuum

text: with

text: the

text: quadrangulation,

text: we

text: have

text: (Σ,

text: V

text: Z

text: and

text: we

text: write

text: a

text: basis

text: over

text: simply

text: as

text: (not

text: V

text: V

text: Bases

text: for

text: graded

text: components.

text: Writing

text: the

text: graded

text: decomposition

text: of

text: (Σ,

text: V

text: as

text: (Σ,

text: V

text: e),

text: we

text: V

text: e

text: then

text: obtain

text: a

text: basis

text: for

text: each

text: (Σ,

text: V

text: e).

text: Let

text: n

text: n

text: denote

text: the

text: number

text: of

text: and

text: respectively

text: among

text: V

text: e

text: e

text: Then

text: (Σ,

text: V

text: e)

text: has

text: basis

text: over

text: [H

text: (Σ)]

text: given

text: by

text: those

text: e

text: e

text: e

text: satisfying

text: I(Σ,V

text: I(Σ,V

text: n

text: n

text: e.

text: These

text: basis

text: elements

text: are

text: precisely

text: those

text: which

text: are

text: suture

text: elements

text: for

text: basic

text: sutures

text: V

text: with

text: Euler

text: class

text: e.

text: It

text: follows

text: that

text: (Σ,

text: V

text: e)

text: is

text: nontrivial

text: precisely

text: for

text: e

text: of

text: the

text: same

text: parity

text: as

text: I(Σ,

text: V

text: and

text: satisfying

text: e

text: I(Σ,

text: V

text: This

text: corresponds

text: to

text: the

text: fact

text: that

text: e(Γ)

text: satisﬁes

text: the

text: same

text: properties,

text: for

text: nontrivial

text: sutures

text: Γ.

text: Sutures

text: up

text: to

text: homeomorphism

text: vs

text: isotopy.

text: Two

text: sets

text: of

text: sutures

text: Γ,

text: Γ

text: on

text: (Σ,

text: V

text: may

text: be

text: homeomorphic

text: in

text: the

text: sense

text: that

text: there

text: is

text: a

text: homeomorphism

text: φ

text: Σ

text: Σ

text: taking

text: Γ

text: to

text: Γ

text: but

text: such

text: Γ,

text: Γ

text: are

text: in

text: general

text: not

text: isotopic.

text: For

text: instance,

text: if

text: Γ

text: is

text: a

text: set

text: of

text: sutures

text: on

text: a

text: disc,

text: and

text: we

text: rotate

text: the

text: disc,

text: the

text: resulting

text: sutures

text: Γ

text: will

text: in

text: general

text: be

text: non-isotopic.

text: Suture

text: elements

text: c(Γ)

text: are

text: invariant

text: under

text: isotopy

text: of

text: sutures,

text: but

text: in

text: general

text: not

text: under

text: homeomorphism;

text: so

text: c(Γ),

text: c(Γ

text: will

text: in

text: general

text: be

text: diﬀerent.

text: D

text: V

text: Keeping

text: track

text: of

text: suture

text: element

text: orbits.

text: We

text: saw

text: in

text: lemma

text: that

text: each

text: (Σ,

text: V

text: V

text: Z

text: Z

text: φ,Γc

text: Z

text: (Σ

text: V

text: sends

text: a

text: [H

text: (Σ)]

text: -orbit

text: into

text: a

text: [H

text: (Σ

text: -orbit.

text: Note,

text: however,

text: that

text: φ∗

text: [H

text: (Σ)]

text: Z

text: D

text: [H

text: (Σ

text: might

text: not

text: be

text: surjective,

text: and

text: so

text: the

text: image

text: of

text: a

text: c(Γ)

text: under

text: might

text: not

text: be

text: an

text: entire

text: Z

text: φ,Γc

text: [H

text: (Σ

text: -orbit.

text: This

text: is

text: why

text: we

text: write

text: c(Γ

text: Γ

text: c(Γ

text: rather

text: than

text: equality.

text: Given

text: the

text: ambiguity

text: D

text: c

text: in

text: however,

text: we

text: may

text: post-multiply

text: by

text: a

text: unit

text: to

text: hit

text: any

text: given

text: element

text: of

text: c(Γ

text: φ,Γc

text: φ,Γc

text: D

text: Gradings

text: of

text: decorated

text: morphisms.

text: Finally,

text: we

text: note

text: that

text: the

text: degree

text: of

text: each

text: is

text: given

text: by

text: the

text: φ,Γc

text: amount

text: by

text: which

text: is

text: shifts

text: Euler

text: classes,

text: i.e.

text: the

text: degree

text: of

text: the

text: decorated

text: morphism

text: deg(φ,

text: Γ

text: c

text: First

text: properties

text: of

text: suture

text: elements

text: We

text: can

text: now

text: develop

text: some

text: basic

text: properties

text: of

text: suture

text: elements.

text: ##LTLine##


text: 3 SUTURED FLOER HOMOLOGY WITH TWISTED COEFFICIENTS by Massot [13] and Ghiggini–Honda [4] for sutured Floer homology. As Ghiggini–Honda note, the extension of sutured Floer homology to the case of twisted coeﬃcients is analogous to the closed case. To explain how twisted coeﬃcients apply to SFH, and extend some results that we will need, we follow closely the arguments of [21] and [20], adapting arguments there to the case of SFH. Roughly, the extra feature in the deﬁnition of SFH with twisted coeﬃcients is that the diﬀerential ∈ ∈ ∂ contains an extra term eA(φ), where A(φ) H (M) is associated to the homotopy class φ π (x, y) 2 2 of a Whitney disc. But to deﬁne A(φ) we ﬁrst need to discuss the homology and homotopy of various spaces involved. 3.4 Homology of the symmetric product We now consider the symmetric product Symk S in some detail, adapting relevant statements of [21, 20] from the closed to the sutured case. There are two maps −→ −→ i : S Symk S, j : H (Symk S) H (S) 1 1 ∈ (cid:55)→ { } deﬁned as follows. Choose a basepoint x S. Then i is deﬁned by x x, x , . . . , x . To deﬁne j 0 0 0 we note that a curve in general position in Symk S gives a map from a k-fold cover of S1 to S; then j is the corresponding map on homology. As a cobordism between curves in general position in Symk S gives a map from a branched cover of a surface to S, j is well-deﬁned. Lemma 3.1 (Cf. Lemma 2.6 of [21]). The maps i and j induce inverse isomorphisms on H . Further, −→ 1 the abelianization map π (Symk S) H (Symk S) is an isomorphism. Thus 1 1 ∼ ∼ H (S) = H (Symk S) = π (Symk S). 1 1 1 Proof. That i and j are inverse on H is clear; hence they are isomorphisms. It remains to show 1 −→ π (Symk S) is abelian; equivalently, any null-homologous γ : S1 Symk S is null-homotopic. Using 1 −→ the map j, a null-homologous γ gives a null-homologous map γˆ : C S, where C is a k-fold cover of −→ | S1. Hence there is a map γ¯ : F S, where F is a 2-manifold with boundary C and γ¯ = γˆ. As in C the proof in [21], increasing the genus of F if necessary, we can assume F is a k-fold branched cover of D −→ D D −→ (cid:55)→ − the disc , with covering map π : F . Then the map Symk S deﬁned by z i(γ¯(π 1(z))) is a null-homotopy of γ. Z In H (S) there are subgroups generated by each [α ] and [β ]. In a balanced sutured Heegaard 1 i i diagram, the curves of α are linearly independent in H (S), and similarly for β, so we have inclusions Z ⊂ 1 of k subgroups H (α), H (β) H (S). 1 1 1T → In Symk S we have the inclusion (cid:44) Symk S, and a subgroup of H (Symk S) given by its image. T α × × · · · × 1 But H ( ) is generated by loops of the form α 0 0, where 0 denotes a constant loop. The 1 α i images of these loops in H (Symk S) are i∗[α ], for i = 1, . . . , k, which are linearly independent since 1 i T → i∗ is an isomorphism and the [α ] are linearly independent in H (S). Thus the inclusion (cid:44) Symk S i T 1 α induces an injective map on H . As both the torus (homeomorphic to (S1)k) and Symk S have 1 α abelian fundamental groups, the inclusion also induces an injective map on π . The same applies for T 1 . We record these results. β Lemma 3.2. There are abelian group inclusions induced by inclusions of spaces ⊂ T T ⊂ T T ⊂ H (α), H (β) H (S), H ( ), H ( ) H (Symk S), π ( ), π ( ) π (Symk S). 1 1 1 1 α 1 β 1 1 α 1 β 1 ∼ ∼ Under the is∼omorpThism∼s H1T(S) ∼= ZH1(Symk S) =∼π1(SyTmk ∼S) indTuced∼bZy i, j and abelianization, we have H (α) = H ( ) = π ( ) = k and H (β) = H ( ) = π ( ) = k. 1 1 α 1 α 1 1 β 1 β × × { } × { } On the other hand, M is obtained by gluing discs to S [0, 1] along α 0 and β 1 , so we have ∼ H (S) ∼ H (Symk S) H1(M) = H (α)1+ H (β) = H (T1 ) + H (T ). 1 1 1 α 1 β 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SUTURED

text: FLOER

text: HOMOLOGY

text: WITH

text: TWISTED

text: COEFFICIENTS

text: by

text: Massot

text: and

text: Ghiggini–Honda

text: for

text: sutured

text: Floer

text: homology.

text: As

text: Ghiggini–Honda

text: note,

text: the

text: extension

text: of

text: sutured

text: Floer

text: homology

text: to

text: the

text: case

text: of

text: twisted

text: coeﬃcients

text: is

text: analogous

text: to

text: the

text: closed

text: case.

text: To

text: explain

text: how

text: twisted

text: coeﬃcients

text: apply

text: to

text: SFH,

text: and

text: extend

text: some

text: results

text: that

text: we

text: will

text: need,

text: we

text: follow

text: closely

text: the

text: arguments

text: of

text: and

text: adapting

text: arguments

text: there

text: to

text: the

text: case

text: of

text: SFH.

text: Roughly,

text: the

text: extra

text: feature

text: in

text: the

text: deﬁnition

text: of

text: SFH

text: with

text: twisted

text: coeﬃcients

text: is

text: that

text: the

text: diﬀerential

text: contains

text: an

text: extra

text: term

text: eA(φ),

text: where

text: A(φ)

text: H

text: (M)

text: is

text: associated

text: to

text: the

text: homotopy

text: class

text: φ

text: π

text: (x,

text: y)

text: of

text: a

text: Whitney

text: disc.

text: But

text: to

text: deﬁne

text: A(φ)

text: we

text: ﬁrst

text: need

text: to

text: discuss

text: the

text: homology

text: and

text: homotopy

text: of

text: various

text: spaces

text: involved.

text: Homology

text: of

text: the

text: symmetric

text: product

text: We

text: now

text: consider

text: the

text: symmetric

text: product

text: Symk

text: S

text: in

text: some

text: detail,

text: adapting

text: relevant

text: statements

text: of

text: from

text: the

text: closed

text: to

text: the

text: sutured

text: case.

text: There

text: are

text: two

text: maps

text: i

text: S

text: Symk

text: S,

text: j

text: H

text: (Symk

text: S)

text: H

text: (S)

text: deﬁned

text: as

text: follows.

text: Choose

text: a

text: basepoint

text: x

text: S.

text: Then

text: i

text: is

text: deﬁned

text: by

text: x

text: x,

text: x

text: x

text: To

text: deﬁne

text: j

text: we

text: note

text: that

text: a

text: curve

text: in

text: general

text: position

text: in

text: Symk

text: S

text: gives

text: a

text: map

text: from

text: a

text: k-fold

text: cover

text: of

text: to

text: S;

text: then

text: j

text: is

text: the

text: corresponding

text: map

text: on

text: homology.

text: As

text: a

text: cobordism

text: between

text: curves

text: in

text: general

text: position

text: in

text: Symk

text: S

text: gives

text: a

text: map

text: from

text: a

text: branched

text: cover

text: of

text: a

text: surface

text: to

text: S,

text: j

text: is

text: well-deﬁned.

text: Lemma

text: (Cf.

text: Lemma

text: of

text: The

text: maps

text: i

text: and

text: j

text: induce

text: inverse

text: isomorphisms

text: on

text: H

text: Further,

text: the

text: abelianization

text: map

text: π

text: (Symk

text: S)

text: H

text: (Symk

text: S)

text: is

text: an

text: isomorphism.

text: Thus

text: H

text: (S)

text: H

text: (Symk

text: S)

text: π

text: (Symk

text: S).

text: Proof.

text: That

text: i

text: and

text: j

text: are

text: inverse

text: on

text: H

text: is

text: clear;

text: hence

text: they

text: are

text: isomorphisms.

text: It

text: remains

text: to

text: show

text: π

text: (Symk

text: S)

text: is

text: abelian;

text: equivalently,

text: any

text: null-homologous

text: γ

text: Symk

text: S

text: is

text: null-homotopic.

text: Using

text: the

text: map

text: j,

text: a

text: null-homologous

text: γ

text: gives

text: a

text: null-homologous

text: map

text: γˆ

text: C

text: S,

text: where

text: C

text: is

text: a

text: k-fold

text: cover

text: of

text: Hence

text: there

text: is

text: a

text: map

text: γ¯

text: F

text: S,

text: where

text: F

text: is

text: a

text: with

text: boundary

text: C

text: and

text: γ¯

text: γˆ.

text: As

text: in

text: C

text: the

text: proof

text: in

text: increasing

text: the

text: genus

text: of

text: F

text: if

text: necessary,

text: we

text: can

text: assume

text: F

text: is

text: a

text: k-fold

text: branched

text: cover

text: of

text: D

text: D

text: D

text: the

text: disc

text: with

text: covering

text: map

text: π

text: F

text: Then

text: the

text: map

text: Symk

text: S

text: deﬁned

text: by

text: z

text: i(γ¯(π

text: is

text: a

text: null-homotopy

text: of

text: γ.

text: Z

text: In

text: H

text: (S)

text: there

text: are

text: subgroups

text: generated

text: by

text: each

text: [α

text: and

text: [β

text: In

text: a

text: balanced

text: sutured

text: Heegaard

text: i

text: i

text: diagram,

text: the

text: curves

text: of

text: α

text: are

text: linearly

text: independent

text: in

text: H

text: (S),

text: and

text: similarly

text: for

text: β,

text: so

text: we

text: have

text: inclusions

text: Z

text: of

text: k

text: subgroups

text: H

text: (α),

text: H

text: (β)

text: H

text: (S).

text: In

text: Symk

text: S

text: we

text: have

text: the

text: inclusion

text: Symk

text: S,

text: and

text: a

text: subgroup

text: of

text: H

text: (Symk

text: S)

text: given

text: by

text: its

text: image.

text: T

text: α

text: But

text: H

text: is

text: generated

text: by

text: loops

text: of

text: the

text: form

text: α

text: where

text: denotes

text: a

text: constant

text: loop.

text: The

text: α

text: i

text: images

text: of

text: these

text: loops

text: in

text: H

text: (Symk

text: S)

text: are

text: i∗[α

text: for

text: i

text: k,

text: which

text: are

text: linearly

text: independent

text: since

text: i

text: T

text: i∗

text: is

text: an

text: isomorphism

text: and

text: the

text: [α

text: are

text: linearly

text: independent

text: in

text: H

text: (S).

text: Thus

text: the

text: inclusion

text: Symk

text: S

text: i

text: T

text: α

text: induces

text: an

text: injective

text: map

text: on

text: H

text: As

text: both

text: the

text: torus

text: (homeomorphic

text: to

text: and

text: Symk

text: S

text: have

text: α

text: abelian

text: fundamental

text: groups,

text: the

text: inclusion

text: also

text: induces

text: an

text: injective

text: map

text: on

text: π

text: The

text: same

text: applies

text: for

text: T

text: We

text: record

text: these

text: results.

text: β

text: Lemma

text: There

text: are

text: abelian

text: group

text: inclusions

text: induced

text: by

text: inclusions

text: of

text: spaces

text: T

text: T

text: T

text: T

text: H

text: (α),

text: H

text: (β)

text: H

text: (S),

text: H

text: H

text: H

text: (Symk

text: S),

text: π

text: π

text: π

text: (Symk

text: S).

text: α

text: β

text: α

text: β

text: Under

text: the

text: is∼omorpThism∼s

text: S)

text: ∼S)

text: indTuced∼bZy

text: i,

text: j

text: and

text: abelianization,

text: we

text: have

text: H

text: (α)

text: H

text: π

text: k

text: and

text: H

text: (β)

text: H

text: π

text: k.

text: α

text: α

text: β

text: β

text: On

text: the

text: other

text: hand,

text: M

text: is

text: obtained

text: by

text: gluing

text: discs

text: to

text: S

text: along

text: α

text: and

text: β

text: so

text: we

text: have

text: H

text: (S)

text: H

text: (Symk

text: S)

text: H

text: H

text: (β)

text: H

text: H

text: (T

text: α

text: β

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text: 3 SUTURED FLOER HOMOLOGY WITH TWISTED COEFFICIENTS P B P A Figure 11: Decomposition of the sutured 3-manifold along U, and its eﬀect on the sutured Heegaard surface: P is split into two copies P and P . A B 32 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

text: SUTURED

text: FLOER

text: HOMOLOGY

text: WITH

text: TWISTED

text: COEFFICIENTS

text: P

text: B

text: P

text: A

text: Figure

text: Decomposition

label: caption

text: of

text: the

text: sutured

text: along

text: U,

text: and

text: its

text: eﬀect

text: on

text: the

text: sutured

text: Heegaard

text: surface:

text: P

text: is

text: split

text: into

text: two

text: copies

text: P

text: and

text: P

text: A

text: B

text: ##LTFigure##

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text: 3 SUTURED FLOER HOMOLOGY WITH TWISTED COEFFICIENTS We can now proceed to the proof of theorem 3.4. Most of the proof in [10] carries over without change. We summarise the proof, detailing the extra steps required for twisted coeﬃcients. Proof. A decomposing surface U satisfying the hypotheses of the theorem can be made good by an isotopy which preserves the set of outer spin-c structures [10, lem. 4.5]. We can then ﬁnd a Heegaard diagram (S, α, β, P) adapted to U [10, prop. 4.4], which is necessarily good, and isotope the α curves (cid:76) to make (S, α, β) admissible, keeping (S, α, β, P) adapted to U [10, prop. 4.8]. Z Denote the subgroups of CF(S, α, βZ; [H2(M)])∼gene⊕rated by oute∼r and inner intersection pZoints by O and I respectively; so CF(S, α, β; [H (M)]) = O I and O = ∈ CF(S, α, β, s; [H (M)]). P P (cid:54) ∅ 2 P P P s OU 2 As π (x, y) = iﬀ s(x) = s(y), the diﬀerential ∂ respects this splitting and we have [10, cor. 5.8] 2 Z | ⊕ | CF(S, α, β; [H (M)]) = (O , ∂ ) (I , ∂ ) . 2 P OP P IP (cid:48) (cid:48) (cid:48) (cid:48)∩ (cid:48) (cid:48) −→ Turning to (S , α , β ), we see that intersection points α β correspond bijectively (via p : S S) ∩ T ∩ T i j to the intersection points α β outside P; so (cid:48) (cid:48) corresponds bijectively to outer intersection T ∩ T i j α(cid:48) (cid:48) β(cid:48) points of . Hence the generators of CF(S , α , β ) (over any coeﬃcient ring) correspond bijec- α β tively to those of O . P Using certain permissible moves (isotopies or handleslides), the diagram (S, α, β, P) can be made nice, while remaining adapted to U and admissible [10, thm. 6.4]; moreover (S, α, β) is also nice [10, (cid:48) (cid:48) (cid:48) prop. 7.5]. The holomorphic curves counted in the diﬀerentials on CF(S , α , β ) (over any coeﬃcient (cid:48) −→ ring) and O correspond bijectively; indeed this bijection is essentially induced by the map p : S S P Z (cid:77) [10, prop. 7.6]. Over coeﬃcients, this implies that (cid:48) (cid:48) Z ∼ (cid:48) (cid:48) (cid:48) Z ∼ | Z ∼ Z SFH(M , Γ ; ) = H(CF(S , α , β ; )) = H(O , ∂ ; ) = SFH(M, Γ, s; ). P OP ∈ s OU (cid:48) (cid:48) (cid:48) Z (cid:48) | ⊂ Over twisted coeﬃcients, however, CF(S , α , β ) has coeﬃcient ring [H (M )], while (O , ∂ ) Z (cid:48) −→ 2 P (cid:48) OP(cid:48) (cid:48) CF(S, α, β) has coeﬃcient ring [H (M)]. Using ι∗ : H (M ) H (M), we may tensor CF(S , α , β ) Z 2 2 2 by [H (M)] to obtain two complexes over the same coeﬃcient ring. The two complexes are then 2 Z free modules over [H (M)], with a bijection between generators induced by p, hence isomorphic as Z 2 [H (M)]-modules: 2 Z ⊗ (cid:48) (cid:48) (cid:48) Z (cid:48) ∼ [H (M)] Z (cid:48) CF(S , α , β ; [H (M )]) = O . 2 [H2(M )] 2 P It remains to show the diﬀerentials are isomorphic. Consider a holomorphic disc with homotopy (cid:48) ∈ (cid:48) (cid:48) (cid:48) (cid:48) ∈ T ∩T (cid:48) (cid:48) (cid:48) Z (cid:48) class φ π (x , y ) contributing to ∂x , where x (cid:48) (cid:48) is a generator of CF(S , α , β ; [H (M )]). (cid:48) 2 ± (cid:48) (cid:48) (cid:48) (cid:48) α(cid:48) (cid:48)β −→ (cid:48) 2 Then ∂x contains a term eA (φ )y , where A : π (x , y ) H (M ) is the assignment described in Z 2 2(cid:48) ± (cid:48) (cid:48) (cid:48) section . After tensoring with [H (M)] using ι∗, the term in ∂x becomes eι∗A (φ )y . 2 The corresponding holomorphic disc contributing to the diﬀerential in O has a homotopy class ∈ (cid:48) (cid:48) (cid:48) −→ P φ π (x, y). Here x = pk(x ), y = pk(y ), where pk : Symk S Symk S is the map naturally 2 (cid:48) −→ ± ± induced by p : S S. Then ∂x contains a corresponding term eA(φ)y with the same sign . Here −→ (cid:48) (cid:48) A : π (x, y) H (M) is a corresponding assignment. It remains to show that ι∗A (φ ) = A(φ). 2 2 (cid:48) (cid:48) (cid:48) −→ (cid:48) Recall from section that (cid:48)A (cid:48): π∼2(x , y(cid:48)) (cid:48) H2(M (cid:48)) ∈is Tdeﬁn∩edT by using a complete set of paths to give an isomorphism π (x , y ) = π (z , z ), where z (cid:48) (cid:48) is some basepoint in the same (cid:48) (cid:48) 2 2 α β spin-c(cid:48) c∩lass as (cid:48)x ∼and y ; (cid:48)then we use the boundarie−s→of a periodic domain to deﬁne an element of H1(α ) H1(β ) = H∼2(M ). Similarly, A : π2(x(cid:48), y) H2(M) uses a complete set of paths for an isomorphism π2(x, y) = π2(z∩, z), wher∼e z = pk(z ), and then takes the boundary of a periodic domain to ﬁnd an element of H (α) H (β) = H (M). 1 1 D2−→ (cid:48) (cid:48) (cid:48) We may compose a Whitney disc u : Symk S from x to y with pk to obtain a Whitney disc ◦ D −→ ∈ (cid:48) (cid:48) pk u : Symk S from x to y. Moreover a homotopy class φ π (x , y ) determines a well-deﬁned ◦ ∈ (cid:48) 2(cid:48) −→ homotopy class pk φ π (x, y). So we obtain a map p∗ : π (x , y ) π (x, y). In fact this map 2 (cid:48) (cid:48) 2(cid:48) Z (cid:48) 2 (cid:48) gives a bijection between the holomorphic discs in CF(S , α , β ; [H (M )]) and O , so p∗(φ ) = φ. 2 P 33 ##LTLine##

text: SUTURED

text: FLOER

text: HOMOLOGY

text: WITH

text: TWISTED

text: COEFFICIENTS

text: We

text: can

text: now

text: proceed

text: to

text: the

text: proof

text: of

text: theorem

text: Most

text: of

text: the

text: proof

text: in

text: carries

text: over

text: without

text: change.

text: We

text: summarise

text: the

text: proof,

text: detailing

text: the

text: extra

text: steps

text: required

text: for

text: twisted

text: coeﬃcients.

text: Proof.

text: A

text: decomposing

text: surface

text: U

text: satisfying

text: the

text: hypotheses

text: of

text: the

text: theorem

text: can

text: be

text: made

text: good

text: by

text: an

text: isotopy

text: which

text: preserves

text: the

text: set

text: of

text: outer

text: spin-c

text: structures

text: lem.

text: We

text: can

text: then

text: ﬁnd

text: a

text: Heegaard

text: diagram

text: (S,

text: α,

text: β,

text: P)

text: adapted

text: to

text: U

text: prop.

text: which

text: is

text: necessarily

text: good,

text: and

text: isotope

text: the

text: α

text: curves

text: to

text: make

text: (S,

text: α,

text: β)

text: admissible,

text: keeping

text: (S,

text: α,

text: β,

text: P)

text: adapted

text: to

text: U

text: prop.

text: Z

text: Denote

text: the

text: subgroups

text: of

text: CF(S,

text: α,

text: βZ;

text: by

text: oute∼r

text: and

text: inner

text: intersection

text: pZoints

text: by

text: O

text: and

text: I

text: respectively;

text: so

text: CF(S,

text: α,

text: β;

text: [H

text: (M)])

text: O

text: I

text: and

text: O

text: CF(S,

text: α,

text: β,

text: s;

text: [H

text: (M)]).

text: P

text: P

text: P

text: P

text: P

text: s

text: OU

text: As

text: π

text: (x,

text: y)

text: iﬀ

text: s(x)

text: s(y),

text: the

text: diﬀerential

text: respects

text: this

text: splitting

text: and

text: we

text: have

text: cor.

text: Z

text: CF(S,

text: α,

text: β;

text: [H

text: (M)])

text: (O

text: (I

text: P

text: OP

text: P

text: IP

text: Turning

text: to

text: (S

text: α

text: β

text: we

text: see

text: that

text: intersection

text: points

text: α

text: β

text: correspond

text: bijectively

text: (via

text: p

text: S

text: S)

text: T

text: T

text: i

text: j

text: to

text: the

text: intersection

text: points

text: α

text: β

text: outside

text: P;

text: so

text: corresponds

text: bijectively

text: to

text: outer

text: intersection

text: T

text: T

text: i

text: j

text: points

text: of

text: Hence

text: the

text: generators

text: of

text: CF(S

text: α

text: β

text: (over

text: any

text: coeﬃcient

text: ring)

text: correspond

text: bijec-

text: α

text: β

text: tively

text: to

text: those

text: of

text: O

text: P

text: Using

text: certain

text: permissible

text: moves

text: (isotopies

text: or

text: handleslides),

text: the

text: diagram

text: (S,

text: α,

text: β,

text: P)

text: can

text: be

text: made

text: nice,

text: while

text: remaining

text: adapted

text: to

text: U

text: and

text: admissible

text: thm.

text: moreover

text: (S,

text: α,

text: β)

text: is

text: also

text: nice

text: prop.

text: The

text: holomorphic

text: curves

text: counted

text: in

text: the

text: diﬀerentials

text: on

text: CF(S

text: α

text: β

text: (over

text: any

text: coeﬃcient

text: ring)

text: and

text: O

text: correspond

text: bijectively;

text: indeed

text: this

text: bijection

text: is

text: essentially

text: induced

text: by

text: the

text: map

text: p

text: S

text: S

text: P

text: Z

text: prop.

text: Over

text: coeﬃcients,

text: this

text: implies

text: that

text: Z

text: Z

text: Z

text: Z

text: SFH(M

text: Γ

text: H(CF(S

text: α

text: β

text: H(O

text: SFH(M,

text: Γ,

text: s;

text: P

text: OP

text: s

text: OU

text: Z

text: Over

text: twisted

text: coeﬃcients,

text: however,

text: CF(S

text: α

text: β

text: has

text: coeﬃcient

text: ring

text: [H

text: (M

text: while

text: (O

text: Z

text: P

text: CF(S,

text: α,

text: β)

text: has

text: coeﬃcient

text: ring

text: [H

text: (M)].

text: Using

text: ι∗

text: H

text: (M

text: H

text: (M),

text: we

text: may

text: tensor

text: CF(S

text: α

text: β

text: Z

text: by

text: [H

text: (M)]

text: to

text: obtain

text: two

text: complexes

text: over

text: the

text: same

text: coeﬃcient

text: ring.

text: The

text: two

text: complexes

text: are

text: then

text: Z

text: free

text: modules

text: over

text: [H

text: (M)],

text: with

text: a

text: bijection

text: between

text: generators

text: induced

text: by

text: p,

text: hence

text: isomorphic

text: as

text: Z

text: [H

text: (M)]-modules:

text: Z

text: Z

text: [H

text: (M)]

text: Z

text: CF(S

text: α

text: β

text: [H

text: (M

text: O

text: P

text: It

text: remains

text: to

text: show

text: the

text: diﬀerentials

text: are

text: isomorphic.

text: Consider

text: a

text: holomorphic

text: disc

text: with

text: homotopy

text: T

text: ∩T

text: Z

text: class

text: φ

text: π

text: (x

text: y

text: contributing

text: to

text: ∂x

text: where

text: x

text: is

text: a

text: generator

text: of

text: CF(S

text: α

text: β

text: [H

text: (M

text: Then

text: ∂x

text: contains

text: a

text: term

text: eA

text: (φ

text: )y

text: where

text: A

text: π

text: (x

text: y

text: H

text: (M

text: is

text: the

text: assignment

text: described

text: in

text: Z

text: section

text: After

text: tensoring

text: with

text: [H

text: (M)]

text: using

text: ι∗,

text: the

text: term

text: in

text: ∂x

text: becomes

text: eι∗A

text: (φ

text: )y

text: The

text: corresponding

text: holomorphic

text: disc

text: contributing

text: to

text: the

text: diﬀerential

text: in

text: O

text: has

text: a

text: homotopy

text: class

text: P

text: φ

text: π

text: (x,

text: y).

text: Here

text: x

text: pk(x

text: y

text: pk(y

text: where

text: pk

text: Symk

text: S

text: Symk

text: S

text: is

text: the

text: map

text: naturally

text: induced

text: by

text: p

text: S

text: S.

text: Then

text: ∂x

text: contains

text: a

text: corresponding

text: term

text: eA(φ)y

text: with

text: the

text: same

text: sign

text: Here

text: A

text: π

text: (x,

text: y)

text: H

text: (M)

text: is

text: a

text: corresponding

text: assignment.

text: It

text: remains

text: to

text: show

text: that

text: ι∗A

text: (φ

text: A(φ).

text: Recall

text: from

text: section

text: that

text: ∈is

text: Tdeﬁn∩edT

text: by

text: using

text: a

text: complete

text: set

text: of

text: paths

text: to

text: give

text: an

text: isomorphism

text: π

text: (x

text: y

text: π

text: (z

text: z

text: where

text: z

text: is

text: some

text: basepoint

text: in

text: the

text: same

text: α

text: β

text: c∩lass

text: as

text: ∼and

text: y

text: we

text: use

text: the

text: boundarie−s→of

text: a

text: periodic

text: domain

text: to

text: deﬁne

text: an

text: element

text: of

text: Similarly,

text: A

text: y)

text: uses

text: a

text: complete

text: set

text: of

text: paths

text: for

text: an

text: isomorphism

text: y)

text: z),

text: wher∼e

text: z

text: pk(z

text: and

text: then

text: takes

text: the

text: boundary

text: of

text: a

text: periodic

text: domain

text: to

text: ﬁnd

text: an

text: element

text: of

text: H

text: (α)

text: H

text: (β)

text: H

text: (M).

text: D2−→

text: We

text: may

text: compose

text: a

text: Whitney

text: disc

text: u

text: Symk

text: S

text: from

text: x

text: to

text: y

text: with

text: pk

text: to

text: obtain

text: a

text: Whitney

text: disc

text: D

text: pk

text: u

text: Symk

text: S

text: from

text: x

text: to

text: y.

text: Moreover

text: a

text: homotopy

text: class

text: φ

text: π

text: (x

text: y

text: determines

text: a

text: well-deﬁned

text: homotopy

text: class

text: pk

text: φ

text: π

text: (x,

text: y).

text: So

text: we

text: obtain

text: a

text: map

text: p∗

text: π

text: (x

text: y

text: π

text: (x,

text: y).

text: In

text: fact

text: this

text: map

text: Z

text: gives

text: a

text: bijection

text: between

text: the

text: holomorphic

text: discs

text: in

text: CF(S

text: α

text: β

text: [H

text: (M

text: and

text: O

text: so

text: p∗(φ

text: φ.

text: P

text: ##LTLine##


text: 4 HEEGAARD DECOMPOSITIONS FOR TWISTED SQFT A B B Figure 19: A Heegaard surface in a Heegaard block. × (cid:48) × { } (cid:48) × { } R−(γ ) S1. The path e 0 connects one sheet to the other. A neighbourhood N(e 0 ) of + (cid:0) (cid:1) + (cid:0) (cid:1) + this path forms a tunnel connecting the two nooks. (cid:3) × Deﬁne two subsets of Σ S1. × \ (cid:48) × { } (i) A = (cid:0)R−(γ ) S1(cid:1) N e(cid:0) 0 .(cid:1) That is, A consists of the bulk, with the tunnel drilled out. + + ⊂ It is a genus 2 handlebody with R− ∂A. × ∪ (cid:48) × { } (ii) B = N(V ) S1 N e 0 . That is, B consists of the two nooks and tunnel joining + + ⊂ them. It is a genus 2 handlebody with R ∂B. + ∩ ∅ ∪ (cid:3) × We have A B = and A¯ B¯ = Σ S1. The Heegaard surface S is their common boundary, ∩ S = ∂A ∂B. We orient S as the boundary of A. Thus S consists of the two sheets, with a 1- (cid:48) × { } (cid:3) × handle or tube attached around e 0 ; it is properly embedded in Σ S1, with oriented boundary × + ∂S = Γ = F S1. It has genus 0 and 4 boundary components. See ﬁgure 19. (cid:48) × To see the α curve of the Heegaard decomposition, consider the annulus e S1 in the bulk region, (cid:48) + lying over the shortened diagon(cid:48)al×e+. I−f the tunnel ha∼s RradiZus r, then this annulus intersects A¯ in a rectangular-shaped disc D = e [r, 2 r] (recall S1 = /2 ), shaded light blue in the diagram. We α + (cid:48) × − set α = ∂D . This α is the union of four closed intervals: two vertical intervals ∂e [r, 2 r], and α (cid:48) × { − } + two horizontal intervals e r, 2 r . + (cid:48) (cid:48)(cid:48) The disc D (green in ﬁgure 20) splits A into two components (A and A ), which are neighbour- ×α (cid:48) ∪ (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) hoods of V− S1. In fact, A deformation retracts onto S D , as does A . Together, A and A ∪ (cid:48) α (cid:48)(cid:48) deformation retract to cover the whole of S D ; A covers one side of D , and A the other. α α The deﬁnition of β is simpler: β a simple closed curve around the tube, bounding a disc D in the β tunnel. In ﬁgure 20 D is shaded in light blue. The disc D blocks the tunnel, splitting B into two (cid:48) β(cid:48)(cid:48) β components (B and B ), which are neighbourhoods of the two nooks. Each component deformation ∪ (cid:48) (cid:48)(cid:48) ∪ (cid:48) retracts onto S D . Together, B and B retract to cover the whole of S D ; B covers one side (cid:48)(cid:48) β β of D , and B covers the other. β (cid:3) × × We observe that Σ S1 is homeomorphic to a thickening S [0, 1] of S, with thickened discs × { } × { } × { } glued to α 0 and β 1 . Under this homeomorphism, R− is given by S 0 surgered along × { } × { } × { } α 0 , and R is given by S 1 surgered along β 1 . So (S, α, β) gives a sutured Heegaard + (cid:3) × × | ∩ | decomposition of (Σ S1, F S1). Note α β = 2. While this construction started with the basic positive sutures γ on the square, we could just as + well perform the same construction starting from γ−. We can then deﬁne a negative Heegaard block, in addition to the positive one described above. For our purposes, however, we only need positive Heegaard blocks as described above. 4.4 Building a Heegaard decomposition from basic sutures with Heegaard blocks Suppose we now have an occupied surface (Σ, V, F) with quadrangulation Q consisting of squares (cid:3) (cid:3) × × (Σ , V , F ). Consider the corresponding balanced sutured 3-manifold with F-sutures (Σ S1, F S1); i i i 41 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: HEEGAARD

text: DECOMPOSITIONS

text: FOR

text: TWISTED

text: SQFT

text: A

text: B

text: B

text: Figure

text: A

text: Heegaard

text: surface

text: in

text: a

text: Heegaard

text: block.

text: R−(γ

text: The

text: path

text: e

text: connects

text: one

text: sheet

text: to

text: the

text: other.

text: A

text: neighbourhood

text: N(e

text: of

label: list

text: this

text: path

text: forms

text: a

text: tunnel

text: connecting

text: the

text: two

text: nooks.

text: Deﬁne

text: two

text: subsets

text: of

text: Σ

text: (i)

text: A

text: N

text: That

text: is,

text: A

text: consists

text: of

text: the

text: bulk,

text: with

text: the

text: tunnel

text: drilled

text: out.

text: It

text: is

text: a

text: genus

text: handlebody

text: with

text: R−

text: ∂A.

text: (ii)

text: B

text: N(V

text: N

text: e

text: That

text: is,

text: B

text: consists

text: of

text: the

text: two

text: nooks

text: and

text: tunnel

text: joining

text: them.

text: It

text: is

text: a

text: genus

text: handlebody

text: with

text: R

text: ∂B.

text: We

text: have

text: A

text: B

text: and

text: A¯

text: B¯

text: Σ

text: The

text: Heegaard

text: surface

text: S

text: is

text: their

text: common

text: boundary,

text: S

text: ∂A

text: ∂B.

text: We

text: orient

text: S

text: as

text: the

text: boundary

text: of

text: A.

text: Thus

text: S

text: consists

text: of

text: the

text: two

text: sheets,

text: with

text: a

text: handle

text: or

text: tube

text: attached

text: around

text: e

text: it

text: is

text: properly

text: embedded

text: in

text: Σ

text: with

text: oriented

text: boundary

text: ∂S

text: Γ

text: F

text: It

text: has

text: genus

text: and

text: boundary

text: components.

text: See

text: ﬁgure

text: To

text: see

text: the

text: α

text: curve

text: of

text: the

text: Heegaard

text: decomposition,

text: consider

text: the

text: annulus

text: e

text: in

text: the

text: bulk

text: region,

text: lying

text: over

text: the

text: shortened

text: I−f

text: the

text: tunnel

text: ha∼s

text: RradiZus

text: r,

text: then

text: this

text: annulus

text: intersects

text: A¯

text: in

text: a

text: rectangular-shaped

text: disc

text: D

text: e

text: [r,

text: r]

text: (recall

text: shaded

text: light

text: blue

text: in

text: the

text: diagram.

text: We

text: α

text: set

text: α

text: ∂D

text: This

text: α

text: is

text: the

text: union

text: of

text: four

text: closed

text: intervals:

text: two

text: vertical

text: intervals

text: ∂e

text: [r,

text: r],

text: and

text: α

text: two

text: horizontal

text: intervals

text: e

text: r,

text: r

text: The

text: disc

text: D

text: (green

text: in

text: ﬁgure

text: splits

text: A

text: into

text: two

text: components

text: (A

text: and

text: A

text: which

text: are

text: neighbour-

text: ×α

text: hoods

text: of

text: V−

text: In

text: fact,

text: A

text: deformation

text: retracts

text: onto

text: S

text: D

text: as

text: does

text: A

text: Together,

text: A

text: and

text: A

text: α

text: deformation

text: retract

text: to

text: cover

text: the

text: whole

text: of

text: S

text: D

text: A

text: covers

text: one

text: side

text: of

text: D

text: and

text: A

text: the

text: other.

text: α

text: α

text: The

text: deﬁnition

text: of

text: β

text: is

text: simpler:

text: β

text: a

text: simple

text: closed

text: curve

text: around

text: the

text: tube,

text: bounding

text: a

text: disc

text: D

text: in

text: the

text: β

text: tunnel.

text: In

text: ﬁgure

text: D

text: is

text: shaded

text: in

text: light

text: blue.

text: The

text: disc

text: D

text: blocks

text: the

text: tunnel,

text: splitting

text: B

text: into

text: two

text: β

text: components

text: (B

text: and

text: B

text: which

text: are

text: neighbourhoods

text: of

text: the

text: two

text: nooks.

text: Each

text: component

text: deformation

text: retracts

text: onto

text: S

text: D

text: Together,

text: B

text: and

text: B

text: retract

text: to

text: cover

text: the

text: whole

text: of

text: S

text: D

text: B

text: covers

text: one

text: side

text: β

text: β

text: of

text: D

text: and

text: B

text: covers

text: the

text: other.

text: β

text: We

text: observe

text: that

text: Σ

text: is

text: homeomorphic

text: to

text: a

text: thickening

text: S

text: of

text: S,

text: with

text: thickened

text: discs

text: glued

text: to

text: α

text: and

text: β

text: Under

text: this

text: homeomorphism,

text: R−

text: is

text: given

text: by

text: S

text: surgered

text: along

text: α

text: and

text: R

text: is

text: given

text: by

text: S

text: surgered

text: along

text: β

text: So

text: (S,

text: α,

text: β)

text: gives

text: a

text: sutured

text: Heegaard

text: decomposition

text: of

text: (Σ

text: F

text: Note

text: α

text: β

text: While

text: this

text: construction

text: started

text: with

text: the

text: basic

text: positive

text: sutures

text: γ

text: on

text: the

text: square,

text: we

text: could

text: just

text: as

text: well

text: perform

text: the

text: same

text: construction

text: starting

text: from

text: γ−.

text: We

text: can

text: then

text: deﬁne

text: a

text: negative

text: Heegaard

text: block,

text: in

text: addition

text: to

text: the

text: positive

text: one

text: described

text: above.

text: For

text: our

text: purposes,

text: however,

text: we

text: only

text: need

text: positive

text: Heegaard

text: blocks

text: as

text: described

text: above.

text: Building

text: a

text: Heegaard

text: decomposition

text: from

text: basic

text: sutures

text: with

text: Heegaard

text: blocks

text: Suppose

text: we

text: now

text: have

text: an

text: occupied

text: surface

text: (Σ,

text: V,

text: F)

text: with

text: quadrangulation

text: Q

text: consisting

text: of

text: squares

text: (Σ

text: V

text: F

text: Consider

text: the

text: corresponding

text: balanced

text: sutured

text: with

text: F-sutures

text: (Σ

text: F

text: i

text: i

text: i

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 TWISTED SQFT Figure 1: Diagonal slides. Standard gluing: Creation: - + + - - + + - - + + + - - + - + - - + + - - + Zip: Fold: - + - + + - + - + - Figure 2: Elementary morphisms. Boundary edges of Σ may be folded or glued under φ; but if they intersect other at endpoints, they coincide. Signs of vertices must be respected. Any homeomorphism of Σ preserving V and V− is a + morphism. Occupied surfaces and morphisms form a category. The empty set is an initial object in this category. In [17] we show any morphism is a composition of the following four types of morphisms, illustrated in ﬁgure 2. (cid:3) (cid:3) (i) A creation takes an occupied surface (Σ, V ) and places a disjoint square (Σ , V ) (the created (cid:48) (cid:48) (cid:116) (cid:3) (cid:3) square) next to it to obtain (Σ , V ) = (Σ, V ) (Σ , V ). (ii) A standard gluing glues two non-consecutive boundary edges of (Σ, V ) respecting orientations, (cid:48) (cid:48) (cid:48) (cid:48) to obtain (Σ , V ) with χ decreased by 1 and N decreased by 1, so I(Σ , V ) = I(Σ, V ). (iii) A fold glues two consecutive boundary edges of (Σ, V ), respecting orientations, which do not (cid:48) (cid:48) (cid:48) comprise an entire boundary component of Σ. The resulting (Σ , V ) has Σ homeomorphic to Σ | (cid:48)| | | − but V = V 2. (iv) A zip glues together two consecutive boundary edges of (Σ, V ) which comprise an entire boundary component. This decreases N by 1 and increases χ by 1. The image of a morphism is a well-deﬁned occupied surface. However, the image occupied surface may have fewer vertices; a vertex of an occupied surface need not be sent to a vertex of the image occupied surface. For instance, this occurs with a fold or zip. In this case we say a vertex has been swallowed. The complement of the image of a morphism is also a well-deﬁned occupied surface. Given a quadrangulated occupied surface, after applying a standard gluing, or creation, we naturally obtain a quadrangulation on the image occupied surface. Even after applying a fold or zip, we obtain a slack quadrangulation of the resulting surface, which is essentially a quadrangulation with vertices permitted in the interior of Σ. Isotoping these interior vertices to boundary vertices we perform slack square collapse and obtain a bona ﬁde quadrangulation. 6 ##LTFigure## ##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##

text: TWISTED

text: SQFT

text: Figure

text: Diagonal

text: slides.

text: Standard

text: gluing:

text: Creation:

text: Zip:

text: Fold:

text: Figure

text: Elementary

text: morphisms.

text: Boundary

text: edges

text: of

text: Σ

text: may

text: be

text: folded

text: or

text: glued

text: under

text: φ;

text: but

text: if

text: they

text: intersect

text: other

text: at

text: endpoints,

text: they

text: coincide.

text: Signs

text: of

text: vertices

text: must

text: be

text: respected.

text: Any

text: homeomorphism

text: of

text: Σ

text: preserving

text: V

text: and

text: V−

text: is

text: a

text: morphism.

text: Occupied

text: surfaces

text: and

text: morphisms

text: form

text: a

text: category.

text: The

text: empty

text: set

text: is

text: an

text: initial

text: object

text: in

text: this

text: category.

text: In

text: we

text: show

text: any

text: morphism

text: is

text: a

text: composition

text: of

text: the

text: following

text: four

text: types

text: of

text: morphisms,

text: illustrated

text: in

text: ﬁgure

text: (i)

text: A

text: creation

text: takes

text: an

text: occupied

text: surface

text: (Σ,

text: V

text: and

text: places

text: a

text: disjoint

text: square

text: (Σ

text: V

text: (the

text: created

text: square)

text: next

text: to

text: it

text: to

text: obtain

text: (Σ

text: V

text: (Σ,

text: V

text: (Σ

text: V

text: (ii)

text: A

text: standard

text: gluing

text: glues

text: two

text: non-consecutive

text: boundary

text: edges

text: of

text: (Σ,

text: V

text: respecting

text: orientations,

text: to

text: obtain

text: (Σ

text: V

text: with

text: χ

text: decreased

text: by

text: and

text: N

text: decreased

text: by

text: so

text: I(Σ

text: V

text: I(Σ,

text: V

text: (iii)

text: A

text: fold

text: glues

text: two

text: consecutive

text: boundary

text: edges

text: of

text: (Σ,

text: V

text: respecting

text: orientations,

text: which

text: do

text: not

text: comprise

text: an

text: entire

text: boundary

text: component

text: of

text: Σ.

text: The

text: resulting

text: (Σ

text: V

text: has

text: Σ

text: homeomorphic

text: to

text: Σ

text: but

text: V

text: V

text: (iv)

text: A

text: zip

text: glues

text: together

text: two

text: consecutive

text: boundary

text: edges

text: of

text: (Σ,

text: V

text: which

text: comprise

text: an

text: entire

text: boundary

text: component.

text: This

text: decreases

text: N

text: by

text: and

text: increases

text: χ

text: by

text: The

text: image

text: of

text: a

text: morphism

text: is

text: a

text: well-deﬁned

text: occupied

text: surface.

text: However,

text: the

text: image

text: occupied

text: surface

text: may

text: have

text: fewer

text: vertices;

text: a

text: vertex

text: of

text: an

text: occupied

text: surface

text: need

text: not

text: be

text: sent

text: to

text: a

text: vertex

text: of

text: the

text: image

text: occupied

text: surface.

text: For

text: instance,

text: this

text: occurs

text: with

text: a

text: fold

text: or

text: zip.

text: In

text: this

text: case

text: we

text: say

text: a

text: vertex

text: has

text: been

text: swallowed.

text: The

text: complement

text: of

text: the

text: image

text: of

text: a

text: morphism

text: is

text: also

text: a

text: well-deﬁned

text: occupied

text: surface.

text: Given

text: a

text: quadrangulated

text: occupied

text: surface,

text: after

text: applying

text: a

text: standard

text: gluing,

text: or

text: creation,

text: we

text: naturally

text: obtain

text: a

text: quadrangulation

text: on

text: the

text: image

text: occupied

text: surface.

text: Even

text: after

text: applying

text: a

text: fold

text: or

text: zip,

text: we

text: obtain

text: a

text: slack

text: quadrangulation

text: of

text: the

text: resulting

text: surface,

text: which

text: is

text: essentially

text: a

text: quadrangulation

text: with

text: vertices

text: permitted

text: in

text: the

text: interior

text: of

text: Σ.

text: Isotoping

text: these

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text: − Superdeformed structures and low parity doublet in Ne S nuclei near neutron drip-line Ω Shailesh K. Singh, S.K. Patra and C.R. Praharaj Institute of Physics, Bhubaneswar-751 005, India (Dated: May 26, 2020) The structure of Ne, Na, Mg, Al, Si, P and S nuclei near the neutron drip-line region is investigated in the frame-work of relativistic mean ﬁeld theory and non-relativistic Skyrme Hartree-Fock formalism. The recently discovered nuclei 40Mg and 42Al, which are beyond the drip-line predicted by various mass formulae are located within these models. We ﬁnd many largely deformed neutron-rich nuclei, whose structures are analyzed. From − the structure anatomy, we ﬁnd that at large deformation, low Ω orbits of opposite parities (e.g. 1+ and 1 ) 2 2 occur close to each other in energy. PACS numbers: 21.10.-k, 21.10.dr, 21.10.Ft, 21.30.-x, 24.10.-1, 24.10.Jv − I. INTRODUCTION the β stability line but also far away from it. Here, we use these two successful models [11–27] to learn about the prop- − erties of drip-line nuclei Ne S. The structure of light nuclei near the neutron drip-line is A. The Skyrme Hartree-Fock (SHF) Method one of the interesting topic for a good number of exotic phe- nomena. Nuclei in this region are quite different in collectivity and clustering features than the stable counterpart in the nu- The general form of the Skyrme effective interaction used clear chart. For example, the neutron magic property is lost in the mean-ﬁeld model can be expressed as a Hamiltonian H H for N = 8 in 12Be [1] and N = 20 in 32Mg [2]. The unexpect- density [13–19]. This is written expressed as a function edly large reaction cross-section for 22C gives the indication of some empirical parameters given as: of neutron halo structure [3]. The discovery of large collectiv- ity of 34Mg by Iwasaki et al. [4] is another example of such H K H H H · · · exotic properties. The deformed structures, core excitation = + 0 + 3 + eff + , (1) and the location of drip-line for Mg and neighboring nuclei are few of the interesting properties of investigation. In this K H H where is the kinetic energy term, the zero range, context, the discovery of 40Mg and 42Al, once predicted to be H 0 3 the density dependent and the effective-mass depen- nuclei beyond the drip-line by various mass formulae [5, 6] eff dent terms, which are relevant for calculating the properties show to need the modiﬁcation of the mass models. of nuclear matter. These are functions of 9 parameters t , x On the other hand, the appearance of N = 16 as magic num- i i (i = 0, 1, 2, 3) a(cid:2)nd η, and are given as (cid:3) ber in 24O and the existence of neutron halo in 11Li are es- tablished observations [7]. However, the proposed proton [8] (8B) and neutron [9, 10] halo (14Be, 17B, 31Ne) in the ex- H = 1t (2 (cid:2)+ x )ρ2 − (2x + 1)(ρ2 + ρ2) , (cid:3) (2) otic nuclei are currently under investigations. In addition to 0 4 0 0 0 p n these, the cluster structure of entire the light mass nuclei and H 1 − = t ρη (2 + x )ρ2 (2x + 1)(ρ2 + ρ2) , (3) the skin formation in neutron-drip isotopes motivate us for the 3 24 3 3 3 p n study of light mass drip-line nuclei. In this paper, our aim is H 1 − = [t (2 + x ) + t (2 + x )] τρ to study the neutron drip-line for Ne S isotopic chain in the eff 8 1 1 2 2 frame-work of a relativistic mean ﬁeld (RMF) and nonrela- 1 − + [t (2x + 1) t (2x + 1)] (τ ρ + τ ρ ()4.) tivistic Skyrme Hartree-Fock (SHF) formalisms and analyzed 8 2 2 1 1 p p n n the features of large quadrupole deformation of these isotopes. The paper is organized as follows: The RMF and SHF for- The kinetic energy K = (cid:126)2 τ, a form used in the Fermi gas malisms are described brieﬂy in Section II. The results ob- model for non-interacting F2eMrmions. The other terms, repre- tained from our calculations are discussed in Section III. Fi- senting the surface contributions of a ﬁnite nucleus with b nally, summary and concluding remarks are given in Section and b(cid:48) as additio(cid:20)nal parameter, are (cid:21) 4 IV. 4 II. THE FORMALISM H 1 (cid:20) 1 − 1 (cid:21) ∇ = 3t (1 + x ) t (1 + x ) ((cid:126) ρ)2 The mean ﬁeld methods like SHF and RMF have been Sρ 16 1 2 1 2 2 2 widely used in the study of binding energies, root mean square − (cid:104)1 1 (cid:105) 1 3t (x + ) + t (x + ) radii, quadrupole deformation and other bulk properties of nu- 16 1 1 2 2 2 2 clei [11, 12]. In general, one can say that although the older × (cid:104)∇ ∇ (cid:105) parametrizations of SHF and RMF have some limitation to ((cid:126) ρn)2 + ((cid:126) ρp)2 , (5) predict the experimental observables, the recent forces are H −1 ∇ · (cid:48) ∇ · ∇ · good enough to reproduce the bulk properties not only near SJ(cid:126) = 2 b4ρ(cid:126) J(cid:126) + b4(ρn(cid:126) J(cid:126)n + ρp(cid:126) J(cid:126)p) . (6) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: title

text: Superdeformed

text: structures

text: and

text: low

text: parity

text: doublet

text: in

text: Ne

text: S

text: nuclei

text: near

text: neutron

text: drip-line

text: Ω

text: Shailesh

label: author

text: K.

text: Singh,

text: S.K.

text: Patra

text: and

text: C.R.

text: Praharaj

text: Institute

text: of

text: Physics,

text: India

text: (Dated:

text: May

label: date

text: The

label: abstract

text: structure

text: of

text: Ne,

text: Na,

text: Mg,

text: Al,

text: Si,

text: P

text: and

text: S

text: nuclei

text: near

text: the

text: neutron

text: drip-line

text: region

text: is

text: investigated

text: in

text: the

text: frame-work

text: of

text: relativistic

text: mean

text: ﬁeld

text: theory

text: and

text: non-relativistic

text: Skyrme

text: Hartree-Fock

text: formalism.

text: The

text: recently

text: discovered

text: nuclei

text: and

text: which

text: are

text: beyond

text: the

text: drip-line

text: predicted

text: by

text: various

text: mass

text: formulae

text: are

text: located

text: within

text: these

text: models.

text: We

text: ﬁnd

text: many

text: largely

text: deformed

text: neutron-rich

text: nuclei,

text: whose

text: structures

text: are

text: analyzed.

text: From

text: the

text: structure

text: anatomy,

text: we

text: ﬁnd

text: that

text: at

text: large

text: deformation,

text: low

text: Ω

text: orbits

text: of

text: opposite

text: parities

text: (e.g.

text: and

text: occur

text: close

text: to

text: each

text: other

text: in

text: energy.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: the

text: β

text: stability

text: line

text: but

text: also

text: far

text: away

text: from

text: it.

text: Here,

text: we

text: use

text: these

text: two

text: successful

text: models

text: to

text: learn

text: about

text: the

text: prop-

text: erties

text: of

text: drip-line

text: nuclei

text: Ne

text: S.

text: The

text: structure

text: of

text: light

text: nuclei

text: near

text: the

text: neutron

text: drip-line

text: is

text: A.

text: The

text: Skyrme

text: Hartree-Fock

text: (SHF)

text: Method

text: one

text: of

text: the

text: interesting

text: topic

text: for

text: a

text: good

text: number

text: of

text: exotic

text: phe-

text: nomena.

text: Nuclei

text: in

text: this

text: region

text: are

text: quite

text: different

text: in

text: collectivity

text: and

text: clustering

text: features

text: than

text: the

text: stable

text: counterpart

text: in

text: the

text: nu-

text: The

text: general

text: form

text: of

text: the

text: Skyrme

text: effective

text: interaction

text: used

text: clear

text: chart.

text: For

text: example,

text: the

text: neutron

text: magic

text: property

text: is

text: lost

text: in

text: the

text: mean-ﬁeld

text: model

text: can

text: be

text: expressed

text: as

text: a

text: Hamiltonian

text: H

text: H

text: for

text: N

text: in

text: and

text: N

text: in

text: The

text: unexpect-

text: density

text: This

text: is

text: written

text: expressed

text: as

text: a

text: function

text: edly

text: large

text: reaction

text: cross-section

text: for

text: gives

text: the

text: indication

text: of

text: some

text: empirical

text: parameters

text: given

text: as:

text: of

text: neutron

text: halo

text: structure

text: The

text: discovery

text: of

text: large

text: collectiv-

text: ity

text: of

text: by

text: Iwasaki

text: et

text: al.

text: is

text: another

text: example

text: of

text: such

text: H

text: K

text: H

text: H

text: H

text: exotic

text: properties.

text: The

text: deformed

text: structures,

text: core

text: excitation

text: eff

text: and

text: the

text: location

text: of

text: drip-line

text: for

text: Mg

text: and

text: neighboring

text: nuclei

text: are

text: few

text: of

text: the

text: interesting

text: properties

text: of

text: investigation.

text: In

text: this

text: K

text: H

text: H

text: where

text: is

text: the

text: kinetic

text: energy

text: term,

text: the

text: zero

text: range,

text: context,

text: the

text: discovery

text: of

text: and

text: once

text: predicted

text: to

text: be

text: H

text: the

text: density

text: dependent

text: and

text: the

text: effective-mass

text: depen-

text: nuclei

text: beyond

text: the

text: drip-line

text: by

text: various

text: mass

text: formulae

text: eff

text: dent

text: terms,

text: which

text: are

text: relevant

text: for

text: calculating

text: the

text: properties

text: show

text: to

text: need

text: the

text: modiﬁcation

text: of

text: the

text: mass

text: models.

text: of

text: nuclear

text: matter.

text: These

text: are

text: functions

text: of

text: parameters

text: t

text: x

text: On

text: the

text: other

text: hand,

text: the

text: appearance

text: of

text: N

text: as

text: magic

text: num-

text: i

text: i

text: (i

text: η,

text: and

text: are

text: given

text: as

text: ber

text: in

text: and

text: the

text: existence

text: of

text: neutron

text: halo

text: in

text: are

text: es-

text: tablished

text: observations

text: However,

text: the

text: proposed

text: proton

text: and

text: neutron

text: halo

text: in

text: the

text: ex-

text: H

text: x

text: otic

text: nuclei

text: are

text: currently

text: under

text: investigations.

text: In

text: addition

text: to

text: p

text: n

text: these,

text: the

text: cluster

text: structure

text: of

text: entire

text: the

text: light

text: mass

text: nuclei

text: and

text: H

text: t

text: ρη

text: x

text: the

text: skin

text: formation

text: in

text: neutron-drip

text: isotopes

text: motivate

text: us

text: for

text: the

text: p

text: n

text: study

text: of

text: light

text: mass

text: drip-line

text: nuclei.

text: In

text: this

text: paper,

text: our

text: aim

text: is

text: H

text: [t

text: x

text: t

text: x

text: τρ

text: to

text: study

text: the

text: neutron

text: drip-line

text: for

text: Ne

text: S

text: isotopic

text: chain

text: in

text: the

text: eff

text: frame-work

text: of

text: a

text: relativistic

text: mean

text: ﬁeld

text: (RMF)

text: and

text: nonrela-

text: [t

text: t

text: (τ

text: ρ

text: τ

text: ρ

text: tivistic

text: Skyrme

text: Hartree-Fock

text: (SHF)

text: formalisms

text: and

text: analyzed

text: p

text: p

text: n

text: n

text: the

text: features

text: of

text: large

text: quadrupole

text: deformation

text: of

text: these

text: isotopes.

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: The

text: RMF

text: and

text: SHF

text: for-

text: The

text: kinetic

text: energy

text: K

text: τ,

text: a

text: form

text: used

text: in

text: the

text: Fermi

text: gas

text: malisms

text: are

text: described

text: brieﬂy

text: in

text: Section

text: II.

text: The

text: results

text: ob-

text: model

text: for

text: non-interacting

text: The

text: other

text: terms,

text: repre-

text: tained

text: from

text: our

text: calculations

text: are

text: discussed

text: in

text: Section

text: III.

text: Fi-

text: senting

text: the

text: surface

text: contributions

text: of

text: a

text: ﬁnite

text: nucleus

text: with

text: b

text: nally,

text: summary

text: and

text: concluding

text: remarks

text: are

text: given

text: in

text: Section

text: and

text: as

text: parameter,

text: are

text: IV.

text: II.

text: THE

text: FORMALISM

text: H

text: x

text: t

text: x

text: The

text: mean

text: ﬁeld

text: methods

text: like

text: SHF

text: and

text: RMF

text: have

text: been

text: Sρ

text: widely

text: used

text: in

text: the

text: study

text: of

text: binding

text: energies,

text: root

text: mean

text: square

text: (x

text: t

text: (x

text: radii,

text: quadrupole

text: deformation

text: and

text: other

text: bulk

text: properties

text: of

text: nu-

text: clei

text: In

text: general,

text: one

text: can

text: say

text: that

text: although

text: the

text: older

text: parametrizations

text: of

text: SHF

text: and

text: RMF

text: have

text: some

text: limitation

text: to

text: predict

text: the

text: experimental

text: observables,

text: the

text: recent

text: forces

text: are

text: H

text: −1

text: good

text: enough

text: to

text: reproduce

text: the

text: bulk

text: properties

text: not

text: only

text: near

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