## Math 524 Introduction to Algebraic Topology, 2018 Fall

### Instructor: Professor Jae Choon Cha

Office hour: by an appointment
Instructor’s web page: http://gt.postech.ac.kr/~jccha/

http://gt.postech.ac.kr/~jccha/algebraic-topology-2018-fall/

### Classroom and hour

Monday and Wednesday 2:00–3:15pm, Science Building II 104

### Syllabus

A brief version is available from POVIS.  A more detailed version is available here.

### Announcement: Final Exam

December 19 Wednesday 2:00pm
Math Building 206

### Homework Problems

Homework problems will be posted on this web page, and will be collected in class on the due date. Turn it in BEFORE the lecture starts.  $\def\C{\mathbb{C}} \def\R{\mathbb{R}} \def\Z{\mathbb{Z}} \def\colim{\operatorname{colim}\limits} \def\Hom{\operatorname{Hom}}$

Problem Set #5 (Due 11/28)

1. Show that $H_n(X;\Z_p)$ is a vector space over the field $\Z_p$. Show that the Euler characteristic of a finite complex $X$ is given by $\chi(X)=\sum_n (-1)^n \dim_{\Z_p} H_n(X;\Z_p)$. What can you say about the case of an arbitrary field $F$ in place of $\Z_p$?
2. Give a detailed proof that a colimit commutes with tensor product: $\colim (A_\alpha\otimes B) \cong (\colim A_\alpha)\otimes B$. (Use that $-\otimes B$ and $\Hom(B,-)$ are adjoint, i.e., $\Hom(A\otimes B,C) \cong \Hom(A,\Hom(B,C))$.)
3. State and give a detailed proof of the following. (You may use what we already proved for chain complexes and homology.)
1. homotopy invariance of cohomology groups,
2. cohomology long exact sequence for a pair,
3. cohomology excision, and
4. equivalence of the singular cohomology and cellular cohomology.
4. Compute the cohomology groups, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$, of the following spaces, using the cellular cochain complex (not using UCT).
1. $S^1\times S^1$
2. Real projective space $\mathbb{R}P^n$
5. [Hatcher, p205] 8 (a) (c)
6. [Hatcher, p205] 9
7. For a Klein bottle $K$, compute the cohomology ring structure on $H^*(K;\Z_2)$ using simplicial cohomology.
8. Give a proof with details that the cochain level cup product induces a well-defined cohomology cup product $H^p(X,A)\times H^q(X,B) \to H^{p+q}(X, A \cup B)$ when both $A$ and $B$ are open subsets of $X$.
9. [Hatcher, p229] 7

#### Problem Set #4 (Due 11/07)

1. [Hatcher, p155] 2
2. [Hatcher, p155] 4
3. [Hatcher, p155] 7
4. [Hatcher, p156] 10
5. [Hatcher, p157] 17
6. Suppose $P(x)=x^n+a_{n-1}x^{n-1} +\cdots a_1 x + a_0$ be a polynomial with complex coefficients, $n>0$. By following the steps below, prove the Fundamental Theorem of Algebra: $P(z)=0$ for some $z\in \C$. In what follows suppose $P(z)\ne 0$ for any $z\in \C$.
1. For $R>0$, let $C_R =\{z\in \C \mid |z|=R\}$.  Show that for all sufficiently large $R>0$, we have $|z^n| > |a_{n-1}z^{n-1} +\cdots + a_1 z + a_0|$ on $C_R$.
2. Show that if $R$ is as in (a), then $P|_{C_R} \colon C_{R} \to \C-\{0\}$ is homotopic to $f\colon C_R \to \C-\{0\}$ defined by $f(z)=z^n$, using a straight line homotopy.
3. Derive a contraction by considering the homomorphisms on $H_1(-)$ induced by $P|_{C_R}$ and $f$.\
7. In what follows, consider the category of abelian groups.
1. Give a detailed proof that a colimit always exists by presenting a construction and checking the required properties rigorously.
2. Show that an element in a direct limit $\varinjlim A_\alpha$ lies in the image of $A_{\alpha_0}\to\varinjlim A_\alpha$ for some $\alpha_0$.
3. Show that an element lies in $\operatorname{Ker}\{A_{\alpha_0}\to\varinjlim A_\alpha\}$ if and only if it lies in $\operatorname{Ker}\{A_{\alpha_0}\to A_\alpha\}$ for some $\alpha\ge\alpha_0$.
8. Find the colimit of each diagram below in the category of abelian groups, and give a proof.
1. $\begin{array}{ccc} A & \to & B \\ \downarrow & & \\ 0 & & \end{array}$
2. $\begin{array}{ccc} 0 & \to & A \\ \downarrow && \\ B && \end{array}$
9. Let $X$ be a CW complex with $k$-skeleton $X^k$.
1. Show that a compact subset of $X$ can meet only finitely many cells.
2. Show $H_i(X)\cong \varinjlim H_i(X^k)$.
3. Using the above, give a detailed proof that the cellular homology of a CW complex $X$ (which is possibly infinite dimensional) is isomorphic to the singular homology, by using a limiting argument.

#### Problem Set #2 (Due 10/22)

Hand in this homework to a box at Math Building 419 (my office) by 2:00pm.

1. Give a fully detailed proof of Snake Lemma.
2. Prove Five Lemma: suppose the following is a commutative diagram of abelian groups with exact rows.
$$\def\l#1{\llap{\scriptstyle#1}}\begin{array}{ccccccccc} A & \to & B & \to & C & \to & D & \to & E \\ \l\alpha\downarrow & & \l\beta\downarrow & & \l\gamma\downarrow & & \l\delta\downarrow & & \l\epsilon\downarrow & & \\ A’ & \to & B’ & \to & C’ & \to & D’ &\to & E’ \end{array}$$
1. If $\alpha$ is surjective and $\beta$, $\delta$ are injective, then $\gamma$ is injective.
2. If $\epsilon$ is injective and $\beta$, $\delta$ are surjective, then $\gamma$ is surjective.
3. Consequently, if $\alpha$, $\beta$, $\delta$, $\epsilon$ are isomorphisms, then $\gamma$ is an isomorphism.
3. For a pair $(X,A)$, give a detailed proof of the reduced version of the homology long exact sequence:
$$\cdots \to \tilde H_i(A) \to \tilde H_i(X) \to H_i(X,A) \to \tilde H_{i-1}(A) \to \cdots$$
4. Suppose $(X,A)$ is a pair. Show the following:
1. If $A$ is contractible, then $\tilde H_n(X) \cong H_n(X,A)$ for any $n$.
2. If $X$ is contractible, then $\tilde H_n(A)\cong H_{n+1}(X,A)$ for any $n$.
5. [Hatcher, p132] 12
6. Prove that the composition of two chain maps is a chain map.
7. Prove that if $f, f’\colon C_* \to D_*$ and $g, g’\colon D_* \to E_*$ are chain maps and $f\simeq f’$, $g\simeq g’$, then $g\circ f \simeq g’\circ f’$.
8. [Hatcher, p132] 14
9. [Hatcher, p132] 16
10. Suppose $C_*$ is a chain complex with each $C_n$ free abelian.  Suppose $C_n=0$ for $n<0$, and $H_n(C_*)=0$ for all $n$. Show that the identity on $C_*$ is chain homotopic to the zero map on $C_*$.
11. Suppose $f\colon X\to Y$ is a map.  Show that $f$ induces a map $Sf\colon SX\to SY$ with the property that $(Sf)_∗\colon \tilde H_{n+1}(SX)\to\tilde H_{n+1}(SY)$ agrees with $f_∗\colon \tilde H_n(X)\to \tilde H_n(Y)$ via the isomorphism $\tilde H_{n+1}(SX)\cong \tilde H_n(X)$.
12. Let $f\colon \partial D^2=S^1 \to S^1$ be the map $z\mapsto z^3$, viewing $S^1$ as the unit circle in the complex plane. Let $X=D^2\cup S^1/\sim$, where $\sim$ is defined by $a\sim f(a)$ for $a\in \partial D^2$. Compute $H_n(X)$. (You may use Mayer-Vietoris.)
13. Suppose $T=S^1\times S^1$ is the torus, and $D\subset T$ is an embedded 2-disk in $T$. Let $X=T-\mathop{int}(D)$.
1. Compute the homology of $X$.
2. Suppose $A$ and $B$ are spaces homeomorphic to $X$ via homeomorphisms $f\colon X\to A$ and $g\colon X\to B$. Denote by $h\colon f(\partial D)\to g(\partial D)$ the composition $g|_{\partial D}\circ (f|_{\partial D})^{−1}$. Compute the homology of the quotient space $Y=(A\cup B)/z∼h(z)$ using Mayer-Vietoris.
14. [Hatcher, p132] 17

#### Problem Set #1 (Due 09/27 23:59)

For this week, hand in your solution to the Math 321 (General Topology, not Math 524!) homework box on the 1st floor of the math building, by the due time.

1. Suppose $X$ is a $\Delta$-complex.
1. Show that $X$ is Hausdorff.
2. Show that $X$ is compact if and only if $X$ has finitely many simplices.
2. Let $X=\Delta^2\times\{0,1\}/(x,0)\sim (x,1)$, $x \in \partial\Delta^2$. Denote the equivalence class of $(x,t)\in \Delta^2\times\{0,1\}$ by $[x,t]\in X$.
1. Show that $X$ is homeomorphic to $S^2$.
2. Define a $\Delta$-complex structure of $X$ for which the map $\sigma_i\colon \Delta^2\to X$ defined by $\sigma_i(x)=[x,i]$ is a characteristic map of a 2-simplex for $i=0,1$. Verify that it is a $\Delta$-complex structure.
3. Consider $\Delta^2=[v_0,v_1,v_2]$. Choose homeomorphisms $f\colon[v_0,v_1]\to[v_1,v_2]$ and $g\colon[v_0,v_1]→[v_2,v_0]$ such that $f(v_0)=v_1$ and $g(v_0)=v_2$. Let $X$ be the quotient space obtained from $\Delta^2$ by the identification $z \sim f(z) \sim g(z)$ for $z\in[v_0,v_1]$.
1. Explain why it is not possible to find a $\Delta$-complex structure of $X$ which has the quotient map $\Delta^2 \to X$ as a characteristic map of a 2-simplex.
2. Find a $\Delta$-complex structure of the space $X$.
4. Compute the simplicial homology groups $H_n^\Delta(X)$ for the $\Delta$-complex $X$ considered in
1. Problem #2
2. Problem #3b
5. Learn what the Klein bottle $K$ is from [Hatcher, p102], and compute $H_i^\Delta(K)$.
6. [Hatcher, p131] #8
7. Suppose $C_*$ and $D_*$ are chain complexes. Define the chain complex $C_*\oplus D_*$ by taking the direct sum of $C_n$ and $D_n$ for each $n$ and the direct sum of the boundary maps. Give a detailed proof that $C_*\oplus D_*$ is a chain complex and $H_n(C_*\oplus D_*)\cong H_n(C_*)\oplus H_n(C_*)$. Generalize this to the case of an infinite sum.
8. Define the kernel and cokernel of a chain map $f\colon C_* \to D_*$ by taking the kernel and cokernel of $f_n\colon C_n\to D_n$ for each $n$.  Show that the kernel and cokernel of $f$, equipped with the induced boundary maps, are chain complexes.