TITLE: When is the connected sum of manifolds orientation-independent?
QUESTION [10 upvotes]: Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed? 
If $N$ has an orientation-reversing automorphism, is this a necessary or sufficient condition for $M$#$N$ to be diffeomorphic to $M$#$\overline{N}$? If it isn't a necessary condition, what invariants can be used to distinguish $M$ # $N$ from $M$ # $\overline{N}$? 
(As a baby example, how does one show that $CP^2$ # $CP^2\ncong CP^2 $#$ \overline{CP ^2}$ ?)
Zygund

REPLY [9 votes]: In the oriented category it's not true that if $M\#N$ is oriented diffeomorphic to $M\#(-N)$ then $N$ admits an orientation reversing diffeomorphism. I suspect there are easier counterexamples but the following works. Take $M^{4k+3}=S^3\times \mathbb{CP}^{2k}$ with $k\ge 1$. Then there exists an exotic sphere $\Sigma^{4k+3}$ such that $\Sigma$ does not admit an orientation reversing diffeomorphism but $M\#\Sigma$ is oriented diffeomorphic to $M$ (and hence also to $M\#(-\Sigma)$). Note that $M$ obviously admits an orientation reversing diffeomorphism because $S^3$ does.
I read about this fact in a paper by Belegradek, Kwasik and Schultz "Codimension two souls and cancellation phenomena". Not sure if this is the earliest reference, perhaps Igor can clarify this - he visits MO regularly.
More specifically they show that if $I(M)$ is the inertia group of $M$ (the group of oriented exotic $4k+3$-spheres $\Sigma$ such that the standard homeomorphis $M\#\Sigma\to M$ is homotopic to a  diffeomorphism)  then $I(S^3\times \mathbb{CP}^{2k})\cap bP_{4(k+1)}$ has index 2 in  $bP_{4(k+1)}$ where $bP_{4(k+1)}$ is the group of exotic $(4k+3)$-spheres bounding parallelizable manifolds. It's known that $bP_{4(k+1)}$ is cyclic or order exponentially growing in $k$ so any nontrivial element $\Sigma$ of $I(S^3\times \mathbb{CP}^{2k})\cap bP_{4(k+1)}$ of order different from 2 works.