This is due March 28.

0. Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it.

A. Let \(A\) be a subspace of \(X\). Show that \(H_0(X,A) = 0\) iff \(A\) meets each path-component of \(X\).

B. 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 \(A\).

C. Let \(A\) be a subspace of \(X\). Show that \(i_\ast: H_n(A) \to H_n(X)\) is an isomorphism for all \(n\) iff \(H_n(X,A) = 0 \) for all \(n\).

D. 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.

E. 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 \(H_n(\mathbb{R}^3, \mathbb{R}^3 – N(K)) \cong H_n(\mathbb{R}^3, \mathbb{R}^3-K)\).

F. 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.