Hatcher p 155

# 3, 4, 8, 9c, 12, 27.

On 9c, describe geometrically the generators of the homology groups.

On 8, follow this hint if you wish: first show that \(\widehat{f}: S^2 \to S^2 \) is homotopic to \(z^{\mathrm{deg\ } f}\) where \( f(z):\mathbb{C}\to\mathbb{C} \) is the polynomial you are given. Then there is only one thing to show on this problem. Why?

Read Example 2.31 p 136, and Exercise 1 p 155.

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.

This will be due March 21.

A. Let \(A\) and \(B\) be abelian groups and \(f\) a homomorphism. Show that \(0\to A\xrightarrow{f} B\) is exact if and only if \(f\) is injective.

B. Suppose \(\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots\) is a long exact sequence. Prove that for each \(n\) there exists a short exact sequence \(0\to\mathrm{coker\ } f_{n+2} \xrightarrow{\phi} A_n \xrightarrow{\theta} \mathrm{ker\ }f_{n-1} \to 0\), where \(\phi\) is induced from \(f_{n+1}\) and \(\theta\) is induced from \(f_n\).

C. Suppose \(A \subset X\), with \(i:A\to X\) the inclusion map, and suppose there is a retraction \(r:X\to A\). Prove that the induced map \(i_\ast: H_n(A)\to H_n(X)\) is a monomorphism and that the induced map \(r_\ast: H_n(X)\to H_n(A)\) is an epimorpishm, for all \(n\).

D. 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 \(n\).

E. Suppose \(A\) is an abelian group and that \(0 \to \mathbb{Z} \xrightarrow{\times 5} \mathbb{Z} \to A \to \mathbb{Z} \to \mathbb{Z} \to 0\) is exact. Classify \(A\).

F. 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 \(\widetilde{H}_0(X)\).

Problems G and H relate to the Snake Lemma. Suppose we have the following commutative diagram of abelian groups where the rows are exact:

\( \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} & & M’ & \ra{f} & M & \ra{g} & M” & \ra{ } & 0 \\ & & \da{d’} & & \da{d} & & \da{d”} & & \\ 0 & \ras{ } & N’ & \ras{f’} & N & \ras{g’} & N” & & \\ \end{array} \)
The Snake Lemma says that the sequence \( \mathrm{ker\ } d’ \xrightarrow{\overline{f}} \mathrm{ker\ } d \xrightarrow{\overline{g}} \mathrm{ker\ } d”\xrightarrow{\partial} \mathrm{coker\ } d’ \xrightarrow{\overline{f’}} \mathrm{coker\ } d \xrightarrow{\overline{g’}} \mathrm{coker\ } d” \) is exact, where \(\partial = (f’)^{-1}\circ d \circ g^{-1}\).

G. Prove the connecting map \(\partial\) is well-defined.

H. Show \(\mathrm{im\ }\partial = \mathrm{ker\ } \overline{f’}\).