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I am reading Kosinski's book. To define the connected sum of $M_1^n$and $M_2^n$ (oriented and closed manifolds) we choose two embeddings of the disk $h_i:\mathbb{D}^n\to M_i$ such that $h_1$ preserves the orientation and $h_2$ reverse it then we can construct the quotient manifold $$\frac{M_1\setminus h_1(0) \sqcup M_2\setminus h_2(0)} \sim $$ where $x\in h_1(\mathbb{D}^n) \sim h_2(\frac{h_1^{-1}(x)}{||h_1^{-1}(x)||^2})\in h_2(\mathbb{D}^n) $ . Now Kosinski shows that this construction doesn't depend on the choice of the embeddings because we know that all the embeddings of $\mathbb{D}^n$ that preserve the orientation are isotopic.

From this though it follows that we can remove the assumption that $h_i$ should preserve(or not) the orientation in the definition of connected sum that shouldn't pose restrictions on the positivity of the diffeomorphism.

I explain why. If say, $h_1$ doesn't preserve the orientation,we just change the orientation on $M_1$ and we get an orientation preserving embedding. Then

if $(-M_1)\sharp M_2$ is diffeomorphic to $M_1\sharp M_2$

we have shown that for the definition it is not necessary to consider orientation. Am I right?

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No, those two manifolds are not always diffeomorphic, or even homotopy equivalent. The simplest counterexample is usually given as $\Bbb{CP}^2 \# \overline{\Bbb{CP}}^2$ and $\Bbb{CP}^2 \# \Bbb{CP}^2$. One has signature 2, the other has signature 0.

If one of the manifolds is not orientable, then there is only one embedding of the disc up to isotopy, and the choice of embedding of the disc in the other manifold doesn't matter.

It is a fluke of luck that you can ignore this for surfaces, where every surface admits an orientation reversing self-diffeomorphism.

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  • $\begingroup$ Do you know of an example of closed orientable (simply connected?) $M$ and $N$ which do not admit orientation reversing diffeos, but yet $M\sharp N$ and $M\sharp \overline{N}$ are diffeomorphic? $\endgroup$ – Jason DeVito Nov 27 '18 at 20:20
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    $\begingroup$ I wasn't aware of the result you just quoted. But here is an example. According to arxiv.org/pdf/1708.06582.pdf (remark 2.8), there is an $1240$-dimensional exotic sphere $\Sigma^{1240}$ of order $7$ for which $\mathbb{H}P^{310}\sharp \Sigma \sharp ...\sharp \Sigma \cong \mathbb{H}P^{310}$ for any number of $\Sigma$ summands. Now, an exotic sphere admits an orientation reversing diffeo iff it's order $2$, so $\Sigma$ works for $N$. Further, $\mathbb{H}P^k$ does not admit an orientation reversing diffeo for $k > 1$ (since $p_1$ is non-trivial), so this works for $M$. $\endgroup$ – Jason DeVito Nov 27 '18 at 20:46
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    $\begingroup$ And, as I'm sure you know, $\overline{\Sigma}\cong \underbrace{\Sigma \sharp ... \sharp \Sigma}_{6\text times}$. Finally, this is the first time in my life that I've used an example whose dimension is in the thousands. Fun day! $\endgroup$ – Jason DeVito Nov 27 '18 at 20:49
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    $\begingroup$ @WarlockofFiretopMountain Once you know the definition of signature, it is immediate from knowing the cohomology ring of the two factors. $\endgroup$ – user98602 Nov 27 '18 at 21:42
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    $\begingroup$ @WarlockofFiretopMountain: Kosinski's book "Differential Manifolds" has a lot of this stuff about connect sums, as well as some of the facts I used about exotic spheres. $\endgroup$ – Jason DeVito Nov 27 '18 at 21:50

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