Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it.
Turn in 3 problems from the ones below
Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. I would recommend (for the midterm, final, and qualifying exams) that you do many other problems. I am available for help.
Due Wednesday, April 1.
Suggested problems:
Hatcher p 131 section 2.1 problems 18, 26.
A. Let \(X\) be a topological space and suppose \(f:X\to X\) is a constant map. Show that the induced map on reduced homology \(f_\ast:\widetilde{H}_n(X)\to \widetilde{H}_n(X) \) is the zero map for all \(n\).
B. Let \(x_0\) be a point in a space \(X\). Show that \(H_0(X,x_0) \cong \widetilde{H}_0(X)\). Hint: play with the quotient \( Z_0(X,x_0) / B_0(X,x_0) \), and recall the theorem involving \(\widetilde{H}_0(X)\).
C. Let \(A\) be a subspace of \(X\). Show that \(H_0(X,A) = 0\) iff \(A\) meets each path-component of \(X\).
D. Let \(A\) be a subspace of \(X\). Show that \(H_1(X,A) = 0\) iff \(H_1(A)\to H_1(X)\) is surjective and each path-component of \(X\) contains at most one path-component of \(A\).
E. Find \(\widetilde{H}_n( S^1\times S^1 / \ast \times S^1 ) \) for all \(n\) using the long exact sequence for the homology of a good pair. Here you might need to use the generators of simplicial homology, but you need not specify generators for the homology groups of the quotient. (you may use the fact that simplicial and singular homology are the same)
F. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow \mathbb{R}^3\). Let \(K = f(S^1\times 0)\) and \(N(K) = f(S^1\times D^2)\). We call \(K\) a knot and \(N(K)\) its tubular neighborhood. We call \(\mathbb{R}^3 – K\) the knot complement. Compute \(H_n(\mathbb{R}^3 – K)\) for all \(n\). Hint: use homotopy invariance and excision, and argue that \(H_n(\mathbb{R}^3, \mathbb{R}^3 – N(K)) \cong H_n(\mathbb{R}^3, \mathbb{R}^3-K)\).
G. Find \(H_n(S^1 \vee (S^1\times S^1) \vee S^2 \vee S^4)\) for all \(n\) and specify generators. Draw a picture.