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) uses::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 homeomorphic to the complex projective plane $$\mathbb{P}^2(\mathbb{C})$$ which has real dimension 4.

There are two possible connected sums:


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