Definition of connected sum and orientation problem

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?

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.
• 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? – Jason DeVito Nov 27 '18 at 20:20
• 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 $M$. – Jason DeVito Nov 27 '18 at 20:46
• 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! – Jason DeVito Nov 27 '18 at 20:49