I am taking a course in homology this semester, and so far we have only examined spaces/surfaces that the simplicial structures are rather easy to find. I was curious about the Alexander Horned Sphere, and how one would approach finding its homology groups.
I figure $H_{0}(X) \cong \mathbb{Z}$, as the horned sphere is path connected (I think), and I know the fundamental group is infinitely generated, but what explicitly would be $H_{1}(X)$?
And then that begs the question of how to find $H_{2} (X)$.
Looking more for a conversation than a proof, just to satisfy my curiosity.
The Alexander horned sphere $A$ is the image of an "exotic" embedding $\phi : S^2 \to \mathbb R^3$. Hence $A \approx S^2$ and $H_n(A) = \mathbb Z$ for $n = 0,2$ and $H_n(A) = 0$ else.