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.

Due February 21.

A. Let \(p:E \to B\) be a regular cover. The group of covering translations \(\mathrm{Aut}(p)\) acts on \(E\). Show that the orbit space of this action is homeomorphic to \(B\), i.e. \(B \cong E/\mathrm{Aut}(p) \).

B. Given a group \(G\) and a normal subgroup \(N\), show that there exists a normal covering space \(E\to B\) with \(\pi_1(B) \cong G\) and \(\pi_1(E) \cong N\), and covering transformation group \(\mathrm{Aut}(p) \cong G/N\).

C. For a path-connected, locally path-connected, and semilocally simply-connected space \(B\), call a path-connected covering space \(E\to B\) abelian if it is normal and has abelian covering transformation group. Show that \(B\) has an abelian covering that is ‘universal’ in the sense that it covers every abelian cover of \(B\). Find the universal abelian cover of \(S^1 \vee S^1\).

D. Construct a two-sheeted covering of the Klein bottle by the torus.

E. Read the section in Hatcher about representing covering transformations by permutations. Let \(B\) be the wedge of two circles, so that \(\pi_1(B) = \langle a, b\rangle\). Construct a three-sheeted cover corresponding to the representation \(\rho: \pi_1(B) \to S_3\) given by \(\rho(a) = (2\,3)\) and \(\rho(b) = (1\,2\,3)\).

I would like your impressions of the course. I have set up an anonymous comment form at 761 Questionnaire. Please answer the three questions and submit the form by February 14.

This will count as a homework assignment. I will ask you to give me your name in class February 14 if you submitted a comment, just to verify that everyone participated.

Due February 14

section 1.3

10, 12, 14, 16

A. Show that the quotient map \(S^2 \to \mathbb{RP}^2 \) is a covering map. This will help on number 14.