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Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two essentially different ways to form the conncected sum $M\#N$. In concrete examples, what methods can be used to show that the two resulting manifolds are not diffeomorphic? Is there an example where the two manifolds are (accidently) diffeomorphic?

Jens Reinhold
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  • Nobody knows something? – Jens Reinhold Feb 03 '15 at 01:39
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    I guess you know http://msp.org/agt/2009/9-4/p18.xhtml where one finds many examples of chiral manifolds. The easiest example to look at is probably $\C P^2\sharp \C P^2$ versus $\C P^2\sharp\overline{\C P^2}$, which should be distinguished by their intersection form. I guess that in the 4-dimensional case, at least for $\pi_1=0$, the intersection form should always suffice to distinguish the two connected sums. – ThiKu Feb 03 '15 at 04:06
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    An answer to you last question can be found here. – Dario Apr 25 '17 at 08:49

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