Let $\mathcal{Man}$ be the set of all connected closed orientable manifolds up to homeomerphism. It is an abelian semigroup with respect to the cartesian product $\times$ (multiplication) and unit $\ast$ (zero dimensional manifold ).

In the same way lets define $\mathcal{Man}^{h}$ as the set of all connected closed orientable manifolds up to homotopy equivalence. It is an abelian semigroup with respect to the cartesian product $\times$ (multiplication) and unit $\ast$ (zero dimensional manifold ).

There is an evident (surjective) homomorphism of abelian semigroups :

$$i:\mathcal{M}\rightarrow \mathcal{M}^{h} $$ $$M\mapsto M $$

which induces a homomorphism of abelian groups after group completion (Grothendieck construction). $$K(i):K(\mathcal{Man})\rightarrow K(\mathcal{Man}^{h}) $$

I do believe that my questions are naive, but let me try to ask them:

1. Is there some (partial) understanding of the groups $K(\mathcal{Man})$ and $ K(\mathcal{Man}^{h})$ ? (e.g. Do they have torsion elements? what are other relations...?)
2. Is there some (partial) understanding of the Kernel of the map $K(i)$ ? (torsion elements, generators, other relations,...)

edit: met me ask a more direct question: suppose that $X$ and $Y$ are connected closed orientable manifolds such that $X\times X$ is homeomorphic to $Y\times Y$, is $X$ homeomorphic to $Y$, or is there a counterexample?