TITLE: Algebraic Topology Homology Group
QUESTION [7 upvotes]: How would one go about calculating the first and second homology groups of a cube inscribed in a sphere?

REPLY [4 votes]: This is too long for a comment, so I'll post it as ananswer. The space ( call it $X$) is clearly homotopy equivalent to two concentric $2$-spheres and $8$ straight line segments connecting them. This space is homeomorphic to $2$ non concentric spheres with $8$ line segments connecting them, which in turn, upon moving the line segments so that they share their starting and ending points on the two spheres, and contracting one of the line segments to a point, is homotopy equivalent to the wedge of two $2$-spheres and seven circles, so that the  space $\mathbf X$ is homotopy equivalent to the wedge sum of two spheres and seven circles. The conclusion is near, since (outside of degree $0$)
$$H_{\bullet}(\bigvee X_i)\simeq\bigoplus H_{\bullet}(X_i).$$
This recovers Jared's result.