Math 74/114
Algebraic Topology
Last updated June 27, 2016 13:25:50 EDT

Announcements:

• Announcements will appear here
• The final exam is due no later than 11am on Thursday, June 6th.
• Turn your exam in in person. I wil have office hours as usual TWTh 2-3:30 as well as Monady, June 3rd, 2-3 and Thursday, June 6th, 10-11.

### Homework Assigments

Week of March 25 to 29, 2013
 Monday: Study: Skim Chapter I.1-6 & I.8. Have a look at I.7 if you like. Do: Suppose that $Y$ and $Z$ are topological spaces and that $q:Y\to X$ is a surjection. Give $X$ the quotient topology. Let $f:X\to Z$ be a function. Define $\bar f:Y\to Z$ by $\bar f(y)=f(q(y))$. Show that $f$ is continuous if and only if $\bar f$ is continuous. Let $X$ be the disjoint union of two copies of the real line $\mathbf{R}$. Formally, $X=\{\,(i,t):\text{$i=1,2$and$t\in \mathbf{R}$}\,\}$ with the appropriate product topology. Let $Y$ be the quotient of $X$ obtained by identifying $(1,t)\sim (2,t)$ if $t > 0$. (An earlier version had $t\ge0$, which is not what I wanted.) Show that $Y$ is locally homeomorphic to $\mathbf{R}$, but that $Y$ is not a $1$-manifold (as it is not Hausdorff). Wednesday: Study: Read sections II.1-3. Do: The Gluing Lemma'' is useful for constructing continuous maps such as homotopies. Suppose that $X$ and $Y$ are spaces and that $X$ is the union of closed sets $A$ and $B$. Suppose that $g:A\to Y$ and $h:B\to Y$ are continuous with $g(x)=h(x)$ for all $x\in A\cap B$. Show that $f:X\to Y$ is continuous when $f(x)=g(x)$ if $x\in A$ and $f(x)=h(x)$ if $x\in B$. Let $\mathsf{C}$ be a category and let $X$ be an object in $\mathsf{C}$. Show that the morphism $1_X\in \operatorname{hom}(X,X)$ is unique. Show that an arcwise connected space $X$ is connected. Conversely, suppose that $X$ is connected and locally arcwise connected. Prove that $X$ is arcwise connected. Friday: Study: Read sections II.4-6. Bonus points if you can find the small gap in Massey's proof of Theorem 5.1. Do: Work problems 4.7, 4.10, 5.3 and 5.4 in the text. (For the last two, you can assume the validity of exercises 5.1 and 5.2.) A category is called "small" if its objects form a set (as opposed to a class). Let $\mathcal{G}(X)$ be the collection of equivalence class of paths in $X$. Explain how to view $\mathcal{G}(X)$ as a small category with objects $X$ such that each morphism is invertible. (One definition of a groupoid is as a small category in which every morphism is invertible.) Let $\phi:X\to Y$ be a continuous map. Observe that $\phi_*$ is a functor from $\mathcal{G}(X)$ to $\mathcal{G}(Y)$. This week's problems will be due Wednesday. Starting from when you read this, homework should be submitted on one side only of normal $8\frac12\times 11$ inch paper. Problems should be clearly marked, and when appropriate, started on a separate page. It is never too soon to learn $\LaTeX$.

Week of April 8 to 12, 2013
 Monday: Study: Read section II.7 and II.8. We'll have very little to say about II.7 in class, but should look at Theorem II.7.1 and its (simple) proof carefully. Also read sections III.1-3. We'll also have little to say about Chapter III in class. Do: Work problems 7.2 and 7.5 (this is identical to 8.2). Recall that a map $f:X\to Y$ is null-homotopic if it is homotopic to a constant map. We say that $X$ is contractible if $\operatorname{id}_X$ is null-homotopic. Note that this is weaker than the definition given on page 45 of the text. (It is the same as Definition VII.4.5 on page168.) Prove that the following are equivalent. $X$ is contractible. Every map $f:X\to Y$ is null homotopic for every $Y$. Every map $g:Y\to X$ is null homotopic for every $Y$. Suppose that $f:X\to Y$ has left homotopy inverse $h:Y\to X$ (i.e., $h\circ f\simeq 1_X$) and a right homotopy inverse $g:X\to Y$ (i.e., $f\circ g\simeq 1_Y$. Show that $f$ is a homotopy equivalence. More generally, show that if $f\circ g$ and $h\circ f$ are homotopy equivalences, then so is $f$. Show that $\mathbf{R}^3\setminus(z\hbox{-axis})$ is homotopy equivalent to $S^1$. Wednesday: Study: Read III.1-6. Do: Let $\mathsf{Ab}$ be the category of abelian groups with group homomorphisms. Show that coproducts exist in $\mathsf{Ab}$ and that the coproduct of $\{G_i,i\in I\}$ is given by the weak product. (In this case, the weak product is usually denoted $\bigoplus_{i\in I} G_i$ and is called the direct sum.) In this problem you can use the facts proved about the weak product proved in sections III.2, and Theorem II.2.1 in particular. Work problems 4.3, 4.5 and 4.8 (assuming 4.7). Work problem 3.3 using the fact that a free abelian group on $S=\{x_i :i\in I\}$ is isomorphic to $\bigoplus_{i\in I} F_i$, where $F_i$ is the infinite cycylic group generated by $x_i$. Thursday (X-HOUR): Study: Read IV.1-2 Do: Give a careful statement of the sense in which Theorem IV.2.1 characterizes $\pi(X)$ up to isomorphism. Work problems IV.3.1 and IV.3.2. Think about IV.3.3 (more on this later). Friday: Study: Read IV.3-5. Do: Work IV.5.4. (Use an appropriate $n$-gon with edge identification as in Example 5.2 in the text). Let $X\subset \mathbf{R}^3$ be the union of $n$ (distinct) lines through the origin. Computer $\pi(\mathbf{R}^3\setminus X)$. Suppose that $X$ and $Y$ are homotopy equivalent and that $X$ is path connected. Must $Y$ be path connected? Explain. (No proofs required.) Let $G$ be a finitely generated abelian group. Find a space $X$ such that $\pi(X)\cong G$. We want to exhibit a path connected compact Hausdorff space $X$ for which $\pi(X)$ is uncountable. Let $C_n$ be the circle in $\mathbf{R}^2$ of radius $\frac 1n$ centered at $(\frac1n,0)$. Let $X=\bigcup_{n=1}^\infty C_n$. Observe that there is a retraction $r^n$ on $X$ onto $C_n$ given by maping each $C_m$, $m\not=n$, to the base point $(0,0)$. Hence there is a homomorphism $r^n_*$ of $\pi(X)$ onto $\pi(C_n)\cong \mathbf{Z}$. Let $J_n=[1-\frac1n,1-\frac1{n+1}]$ and let $h_n:J_n\to I$ be the unique orientation preserving linear homeomorphism. Let $f_n:I\to C_n$ be any path in $C_n$. Define $f:I\to X$ by letting $f|_{J_n}=f_n \circ h_n$, and $f(1)=(0,0)$. Show that $f$ is a path in $X$. (Your proof may include the words "$f$ is clealy continuous at any $t\not=1$".) Show that there is a homomorphism $p$ of $\pi(X)$ onto $\prod_{i=1}^\infty \mathbf{Z}$. (Since the latter is uncountable, this means $\pi(X)$ is as well.) Does this result contradict the result from problem IV.3.2 which shows that the fundamental group of a countable wedge product of circles is the free group, $\mathbb{F}_\infty$, on countably many generators (which is countable)? Explain.

Week of April 8 to 12, 2013
 Monday: Study: Read sections V.1-3. Note that Massey uses the phrase "$f$ maps $V$ topologically onto $U$" to me that $f$ restricts to a homemorphism of $U$ onto $V$. Do: Establish the following assertion made in class: if $(\tilde X,p)$ is a convering space for $X$ and if $V$ is a proper, open, path connected subset of $\tilde X$ such that $p(V)=X$, then $(V,p|_V)$ is not a covering space for $X$ even though $p|_V$ is a local homeomorphism of $V$ onto $X$. Work problems V.2.3 and V.2.4. (Note that Massey is making no assumptions on topological spaces other than that there are path connected and locally path connected. In particular, they need not be metric spaces and the topology need not be determiend by sequences. But if you need to resort to sequences you can, but just note that there is a "sequence free" solution out there.) Let $(\tilde X,p)$ be covering space for $X$. A open path connected subset $U$ of $\tilde X$ is called fundamental if $p(U)$ is elementary and $U$ is a path component of $p^{-1}(p(U))$. Show that the topology on $\tilde X$ has a basis of fundamental sets. (Hint: this is more or less what Massey is proving on the top of page 129.) Wednesday: Study: Read V.4-5 Do: Work problems V.5.1, V.5.2 and V.5.3. Thursday (X-HOUR): Study: Read V.6 Do: Work problem V.6.1 (Define a surjection from $\pi(X,x_0)$ to $p^{-1}(x_0)$.) Friday: Study: Read V.7 Do: Work problems 6.2 and 7.1.

Week of April 15 - 19, 2013
 Monday: Study: Read V.8-9. For your own edification, you ought to skip V.10 and especially Lemma V.10.1. Do: Let $G$ be a group of homeomorphisms of a space $X$. Prove that the oribit map $p:X\to X/G$ is open. Work problem V.9.1. Work problem V.9.2 and conclude that $\mathbf{R}^n$ is never homeomorphic to $\mathbf{R}^{n+1}$ (for $n\ge1$). Wednesday: Study: Read VII.1-2. Also, looking at all of Chapter VI would be very profitable. It will at least provide some motivation for the definitions to come. Do: Show that $\partial_{n-1}\circ \partial_n = 0$. Frankly, this is a better test of exposition than anything else. You may want to prove a lemma using the following observation: $\sum_{i=1}^n\sum_{j=1}^{n-1} c_{ij} =\sum_{i=2}^n\sum_{j=1}^{i-1} c_{ij} + \sum_{i=1}^{n-1} c_{ii} + \sum_{j=2}^{n-1}\sum_{i=1}^{j-1} c_{ij}.$ This just amounts to breaking the sum up into the bits where $i>j$, $i=j$ and $j>i$. Friday: Study: Read VII.3-4 Do: Work problem VII.3.3. Also skim X1.1-2. A Chain complex $C^\bullet$ is a family $\{\,C_n:n\in\mathbf{Z}\,\}$ of abelian groups together with homomorphisms $\partial_n:C_n\to C_{n-1}$ such that $\partial_{n-1}\circ \partial_n=0$. (An example is given by the groups $C_n(X)$ and boundary maps $\partial_n$ constructed in Section 2.1 and 2.2 of Chapter VII or our text.) A chain map from $C^\bullet$ to $D^\bullet$ is a collection of homomorphisms $f_n:C_n\to D_n$ such that $\partial^D_n\circ f_n = f_{n-1}\circ \partial^C_n$ for all $n$. (Draw the appropriate commutative diagram.) Describe how to define a funtor $H_n(\cdot)$ from the cateory of chaim complexes and chain maps to the category of abelian groups so that $H_n(C^\bullet)$ is $H_n(X)$ when $C^\bullet$ is the chain complex associated to $X$ as above. If $f,g:C^\bullet \to D^\bullet$ are chain maps, then homomorphisms $d_n:C_n \to D_{n+1}$ are called a chain homotopy between $f$ and $g$ if $f_n-g_n = \partial^D_{n+1}\circ d_n + d_{n-1}\circ \partial_n$. Show that in this case, $f_*=g_*:H_n(C^\bullet)\to H_n(D^\bullet)$.

Week of April 22 to 26, 2013
 Monday: Study: Read VII.4-5 Do: No Assignments to turn in this week. Take-home is due Wednesday, May 1st. Wednesday: Study: NO CLASS TODAY Do: Friday: Study: VII.5-6. We are not covering the proof of Theorem VII.6.4 (which is all of section VII.7). Do: Take-home exam is due WEDNESDAY, May 1st. Recommended: Fill in the details of the proofs of Theorem VII.5.1 and Proposition VII. Work VII.5.1. Observe that the integers $\mathbf{Z}$ and the rationals $\mathbf{Q}$ (with their usual topologies viewed as subsets of $\mathbf{R}$) satisfy $H_n(\mathbf{Z})\cong H_n(\mathbf Q)$ for all $n\ge 0$. Are $\mathbf Z$ and $\mathbf Q$ homotopy equivalent?

Week of April 29 to May 3, 2013
 Monday: Study:Read VIII.1-2 Do: Work problem VII.5.2. Work problem VII.5.5. Wednesday: Study: Read VIII.3 Do: Work VIII.2.3, VIII.2.4, VIII.2.5, VIII.2.7, VIII.2.9. Prove the snake lemma. For a hint on the snake-lemma, and why it's called the "snake lemma", see this clip of Jill Clayburgh in "It's My Turn". Note that we are meeting in our x-hour (Thursday) this week and next. :-( Thursday (X-HOUR): Study:Still working on VIII.3 Do: Complete the proof of Theorem VIII.3.4 in the text. Prove the following assertion used in lecture. Let $f:A\to F$ be an epimorphism of free abelian groups so that $\ker f$ is a (necessarily free abelain group) of rank $k$. Suppose that $S:=\{z_1,z_2,\dots,z_k,e_1,\dots,e_n\}$ generates $A$ and that $A$ has rank $k+n$. (Hence $S$ is a basis for $A$.) Prove that $\{z_1 ,z_2 ,\dots,z_k \}$ is a basis for $\ker f$. Friday: Study: Read VIII.4 Do: Work problems VIII.3.2-6. Work this problem on natural transformations.

Week of May 6 to 10, 2013
 Monday: Study: Go over these solutions to exam one and compare with your solutions. Read section VIII.5 and X.1-2 as necessary. Do: Complete the proof of Theorem VIII.5.1 dealing with reduced homotopy (in dimension 0) and naturality. Work problems VIII.5.1 and VIII.5.2. Wednesday: Study: Read VIII.6 Do: No assignment today Optional: Using these notes on direct limits of groups, show that $H_q(X,A)$ is the direct limit of all the $i_*^{(C,D)}(H_q(C,D))$ for $(C,D)$ ranging over all compact sub-pairs $(C,D)$ of $(X,A)$. (Here $i^{(C,D)}$ is the inclusion map of $(C,D)$ into $(X,A)$.) Thursday (X-HOUR): Study:Reread VIII.6 Do: Work VIII.6.1-4, VIII.6.7 and VIII.6.8. Movie:Koch snowflake movie. Friday: Study: Skim VIII.7 for your own edification. Read IX.1-2. Do: If $\mathcal F$ is a family of functions $f:X_f\to Y$ and if each $X_f$ has a topology $\tau_f$, then there is a largest topology $\tau$ on $Y$, called the final topology induced by $\mathcal F$ (or in the text, the weak topology induced by $\mathcal F$) such that each $f\in \mathcal F$ is continuous. Show that $\tau=\{\,U\subset Y:\text{$f^{-1}(U)\in \tau_f$for all$f\in \mathcal F$}\,\}$. Let $Y$ have the final topology $\tau$ as above. Show that $g:Y\to Z$ is continuous if and only if $g\circ f$ is continuous for all $f\in\mathcal F$.
 Monday: Study: Read section IX.4 Do: Prove that $\{ C_n(K^\bullet),d_n\}$ is a chain complex; that is, show that $d_{n-1}\circ d_n=0$. Prove assertions (a) and (b) about finitely generated abelian groups on the top of page 236 of the text. Note that finite abelian groups $F$ satisfy $r(F)=0$. Prove Lemma IV.4.3 on page 236 of the text. Wednesday: Study: Read X.3 Do: Work X.2.6. Suppose that $f:A\to C$ and $g:B\to D$ are homomorphisms between abelian groups. Show that there is a unique homomorphism $f\otimes g:A\otimes B\to C\otimes D$ such that $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$. Prove the following assertion about short exact sequences from lecture. Friday: Study: For your own edification, you should carefully read X.4. Also read X.5. Do: Let $G$ be an abelian group. Using the computation of $H_q(\{pt\};G)$ on page 265 of the text, the appropriate analogue of Proposition VII.2.7 and the Mayer-Veitoris sequence, to compute $H_q(S^n;G)$. (See Example VIII.5.1.) A chain complex $K_\bullet$ is called acyclic if $H_q(K_\bullet)=0$ for all $q$. Suppose that $K_\bullet$ and $L_\bullet$ are acyclic chain complexes with $K_0=L_0=\{0\}$ for all $n<0$, and that $K_n$ is free abelian for all $n>0$. Let $\phi:K_0\to L_0$ be a homomorphism. Show that there is chain map $\tau:K_\bullet \to L_\bullet$ such that $\tau_0=\phi$, and that $\tau$ is unique up to chain homotopy. To define $\tau_n$ for $n>0$, note that if $\{s_i\}$ is a basis for $K_n$, then it suffies to define $\tau(s_i)$. Then notice that $\tau_{n-1}\circ \partial_n\in \ker \partial'_{q-1}$. Then you need to define $d_n:K_n\to L_{n+1}$ such that $\partial'_{n+1}\circ d_n=\tau_n-\tau'_n-d_{n-1}\circ \partial_n$. But the right-hand side is in $\ker \partial'_n$. Please Take Note: This is the last homework assignment. As usual, it is "due" Wednesday in my mailbox. However, the earliest that I will pick it up is Monday the 27th. I will pass out the final on Wednesday, May 29th (also known as the last day of classes). The final is due on Monday, June 3rd, from 2-3:30 in my office. Anyone who asks can get an extension up to Thursday Morning, June 6th.