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/

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.

Homework Problems

Homework problems will be posted on this web page every other week, 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 #1 (Due 09/27)

  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.