Homework 13

Due Thursday, April 25, in class.

A. Prove the splitting lemma.

p. 176, Hatcher

# 1, 2, 8

Extra hint for 2: it is easy to get confused about the subscripts. Recall that a connected graph has (reduced) nonzero homology only in dimension one. This observation will save you a lot of trouble when considering Mayer-Vietoris sequences. Once you have figured out the case where \(X\) is a tree, build up to arbitrary graphs by adding edges one at a time.

Homework 12

Due in class April 18.

p 155, section 2.2: 21, 22, 28a, 29

A. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow S^3\). 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 \(S^3 – K\) the knot complement. Compute \(H_n(S^3 – K)\) for all \(n\) using Mayer-Vietoris. Carefully specify generators of all spaces involved.

B. Consider \(\mathbb{R}^3 – K \subset S^3 – K \). Using a Mayer-Vietoris sequence, find a generator for \(H_2(\mathbb{R}^3-K)\).

C. Write a short essay outlining a proof that for a homology theory \(h\) on the category of finite CW pairs, the group \(h_0(\ast)\) determines all homology groups \(h_n(X,A)\) for all CW pairs \((X,A)\). A flow chart may be useful.

Homework 11

This assignment is due April 11 in class.

A. Let \(M_2 \) denote the orientable genus two surface, and let \(X\) denote \(M_2 – \mathrm{int\ } D^2\). Call the boundary curve \(\gamma\). Form a space \(Y\) by attaching a disk \(D^2\) along its boundary to the curve \(\gamma\) by a map of degree \(5\). Compute the homology groups of \(Y\).

B. Let \(X = S^1 \vee S^1\), the circles labeled \(a\) and \(b\), and form \(Y\) by gluing two 2-cells onto \(X\) by the identifications \(a^4\) and \(a^4b^{-2}a^2b^2a^{-2}\). Compute the homology groups of \(Y\).

C. Recall that \(\mathbb{R}P^3 = e^0\cup e^1 \cup e^2 \cup e^3\). Let \(X\) be the space obtained by attaching a 4-cell to \(\mathbb{R}P^3\) where the composition of the quotient map and attaching map \(\Delta: S^3 \xrightarrow{\phi} \mathbb{R}P^3 \xrightarrow{q} \overline{e^3}/\partial{e^3} \cong S^3\) has degree \(3\). Compute the homology groups of \(X\).

D. Recall that \(\mathbb{R}P^3\) is naturally a subcomplex of \(\mathbb{R}P^4\). Compute the homology of \(\mathbb{R}P^4/ \mathbb{R}P^3\) using cellular homology.

Homework 10

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.

Homework 9

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.

Homework 8

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

Homework 7

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.

Homework 6

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 \(\rho(b) = (1\,2\,3)\).

Homework 5

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.