Homotopy type of connected sum depends on choice of gluing map

Statement

It is possible to find an example of compact connected orientable manifolds $M_1$ and $M_2$ such that the homotopy type of the connected sum $M_1 \# M_2$ is not well defined, i.e., we can get connected sums of different homotopy types depending on the choice of the gluing map.

Facts used

1. Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension

Proof

To construct an example, we need to find a case where both $M_1$ and $M_2$ are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both $M_1$ and $M_2$ as isomorphic to the complex projective plane $\mathbb{P}^2(\mathbb{C})$.

There are two possible connected sums:

• Connected sum of two complex projective planes with same orientation: This has cohomology ring isomorphic to $\mathbb{Z}[x,y]/(x^2 - y^2,x^3,y^3)$, where $x,y$ are additive generators of the free abelian group $H^2$ and $x^2 = y^2$ is the additive generator for $H^4$.
• Connected sum of two complex projective planes with opposite orientation: This has cohomology ring isomorphic to $\mathbb{Z}[x,y]/(x^2 + y^2,x^3,y^3)$, where $x,y$ are additive generators of the free abelian group $H^2$ and $x^2 = -y^2$ is the additive generator for $H^4$.