This is a two week assignment, due March 7. This assignment is finally complete.

A. Consider the sequence of homomorphisms \(0 \to \mathbb{Z}^2 \xrightarrow{g} \mathbb{Z}^4 \xrightarrow{f} \mathbb{Z}^3 \to 0 \), where in the standard bases of \(\mathbb{Z}^n\), \(g = \begin{pmatrix} 2 & 0 \\ 0 & 2 \\ -4 & 8 \\ 2 & -4 \end{pmatrix}\) and \(f = \begin{pmatrix} -3 & 6 & -1 & 1\\ 1 & -2 & 2 & 3 \\ 2 & -4 & 5 & 8 \end{pmatrix} \). Verify that this is a chain complex, and compute the homology groups at each level. Specify generators for the homology groups in terms of the standard bases for the \(\mathbb{Z}^n\)’s in terms of which \(g\) and \(f\) are given.

B. Let \(X\) be a point. Find the simplicial homology groups \(H_n^\Delta(X)\), for \(n\geq 0\).

C. Let \(X\) be the Klein bottle. Use the \(\Delta\)-complex structure on page 102 of Hatcher to compute \(H_n^\Delta(X)\), for \(n\geq 0\). Identify the generators of the homology groups with linear combinations of simplices.

D. Recall that for a covering space \(p: E \to B\), the map induced on the fundamental groups is a monomorphism. In light of Hurewicz’ Theorem, there is a map induced on first homology. Does the map on homology need to be a monomorphism?

E. Hatcher Section 2.1 number 8 (page 131). You will get 3/5 points for picking \(n=5\), and full points for the general case.