### Instructor: Professor Jae Choon Cha

Office hour: by an appointment

Instructor’s web page: http://gt.postech.ac.kr/~jccha/

### Course home page

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)**

- 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 $\Z_p$?
- 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., $\Hom(A\otimes B,C) \cong \Hom(A,\Hom(B,C))$.)
- State and give a detailed proof of the following. (You may use what we already proved for chain complexes and homology.)
- homotopy invariance of cohomology groups,
- cohomology long exact sequence for a pair,
- cohomology excision, and
- equivalence of the singular cohomology and cellular cohomology.
- 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).
- $S^1\times S^1$
- Real projective space $\mathbb{R}P^n$
- [Hatcher, p205] 8 (a) (c)
- [Hatcher, p205] 9
- For a Klein bottle $K$, compute the cohomology ring structure on $H^*(K;\Z_2)$ using simplicial cohomology.
- 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 $X$.
- [Hatcher, p229] 7

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

- [Hatcher, p155] 2
- [Hatcher, p155] 4
- [Hatcher, p155] 7
- [Hatcher, p156] 10
- [Hatcher, p157] 17
- 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 $z\in \C$. - 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 $C_R$.
- 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.
- Derive a contraction by considering the homomorphisms on $H_1(-)$ induced by $P|_{C_R}$ and $f$.\
- In what follows, consider the category of abelian groups.
- Give a detailed proof that a colimit always exists by presenting a construction and checking the required properties rigorously.
- 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 $\alpha_0$.
- 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 $\alpha\ge\alpha_0$.
- Find the colimit of each diagram below in the category of abelian groups, and give a proof.
- $\begin{array}{ccc} A & \to & B \\ \downarrow & & \\ 0 & & \end{array}$
- $\begin{array}{ccc} 0 & \to & A \\ \downarrow && \\ B && \end{array}$
- Let $X$ be a CW complex with $k$-skeleton $X^k$.
- Show that a compact subset of $X$ can meet only finitely many cells.
- Show $H_i(X)\cong \varinjlim H_i(X^k)$.
- 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.

- Give a fully detailed proof of Snake Lemma.
- 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}

$$ - If $\alpha$ is surjective and $\beta$, $\delta$ are injective, then $\gamma$ is injective.
- If $\epsilon$ is injective and $\beta$, $\delta$ are surjective, then $\gamma$ is surjective.
- Consequently, if $\alpha$, $\beta$, $\delta$, $\epsilon$ are isomorphisms, then $\gamma$ is an isomorphism.
- For a pair $(X,A)$, give a detailed proof of the reduced version of the homology long exact sequence:

$$ \cdots \to \tilde H_i(A) \to \tilde H_i(X) \to H_i(X,A) \to \tilde H_{i-1}(A) \to \cdots$$ - Suppose $(X,A)$ is a pair. Show the following:
- If $A$ is contractible, then $\tilde H_n(X) \cong H_n(X,A)$ for any $n$.
- If $X$ is contractible, then $\tilde H_n(A)\cong H_{n+1}(X,A)$ for any $n$.
- [Hatcher, p132] 12
- Prove that the composition of two chain maps is a chain map.
- 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 $g\circ f \simeq g’\circ f’$.
- [Hatcher, p132] 14
- [Hatcher, p132] 16
- 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 $C_*$.
- 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 $\tilde H_{n+1}(SX)\cong \tilde H_n(X)$.
- 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.)
- Suppose $T=S^1\times S^1$ is the torus, and $D\subset T$ is an embedded 2-disk in $T$. Let $X=T-\mathop{int}(D)$.
- Compute the homology of $X$.
- 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.
- [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.

- Suppose $X$ is a $\Delta$-complex.
- Show that $X$ is Hausdorff.
- Show that $X$ is compact if and only if $X$ has finitely many simplices.
- 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 $[x,t]\in X$.
- Show that $X$ is homeomorphic to $S^2$.
- 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.
- 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 $z\in[v_0,v_1]$.
- 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.
- Find a $\Delta$-complex structure of the space $X$.
- Compute the simplicial homology groups $H_n^\Delta(X)$ for the $\Delta$-complex $X$ considered in
- Problem #2
- Problem #3b
- Learn what the Klein bottle $K$ is from [Hatcher, p102], and compute $H_i^\Delta(K)$.
- [Hatcher, p131] #8
- 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.
- 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.