1. Categories and Chain Complexes (31/1/2022).
  2. Simplicial Complexes and Simplicial Homology (7/2/2022).
  3. CW-Complexes (14/2/2022).
  4. Homology : Overview (21/2/2022).

Categories and Chain Complexes

due by Monday, Jan 31, 2022

Recall that we can associate to a partially ordered set $(X, \leq)$ a category with objects elements of $X$ and morphisms corresponding to pairs $A, B\in X$ with $A \leq B$, with the morphism corresponding to $(A, B)$ having domain $A$ and codomain $B$.

  1. Given partially ordered sets $(X, \leq)$ and $(Y, \leq)$, show that there is a bijection between order-preserving functions from $X$ to $Y$ and functors from the category corresponding to $X$ to the category corresponding to $Y$.

In what follows, by a chain complexes $(C_*, \del_*)$ we mean a sequence of abelian groups $\{C_n\}_{n\geq 0}$ for $n\geq 0$ and homomorphisms $\{\del_n : C_n \to C_{n-1}\}_{n\geq 1}$, such that $\del_n\circ \del_{n+1} = 0$ for $n\geq 0$.

  1. Let $(C_n, \del_n)$ and $(C'_n, \del'_n)$ be two chain complexes. Let $\{H_n: C_n \to C'_{n+1}\}_{n\geq 0}$ be a family of homomorphisms. Define $\varphi_n = \del'_{n + 1}\circ H_n + H_{n - 1} \circ \del_n$ for $n\geq 1$ and $\varphi_0 = \del'_1\circ H_0.$ Show that $\{\varphi_n\}_{n\geq 0}$ is a chain homomorphism.
  2. Let $(C_n, \del_n)$, $(C'_n, \del'_n)$ and $(C''_n, \del''_n)$ be chain complexes. Let $\varphi, \psi: C_* \to C'_*$ and $\varphi', \psi': C'_* \to C''_*$ be chain homomorphisms. Suppose $\varphi$ is chain homotopic to $\psi$ and $\varphi'$ is chain homotopic to $\psi'$. Show that $\varphi'\circ\varphi$ is chain homotopic to $\psi'\circ\psi$.
  3. Let $(C_n, \del_n)$ be the chain complex with $C_n = \Z$ for all $n\geq 0$, with boundary homomorphisms given by, for $k\geq 0$, $$\del_{2k + 1}:\Z\to \Z = 0, \\ \del_{2k + 2}:\Z\to \Z =\unicode{x1D7D9}_{\Z},$$ i.e., the $0$ homomorphism in odd degrees and the identity homomorphism in even (non-zero) degrees. Let $\varphi_*: C_*\to C_*$ be the chain homomorphism with $\varphi_0 = \unicode{x1D7D9}_{\Z}$ (the identity) and $\varphi_n = 0$ (the zero homomorphism) for $n \geq 1$.
    • (a) Show that $\varphi$ is chain homotopic to the identity chain homomorphism $\unicode{x1D7D9}: C_* \to C_*$.
    • (b) Deduce that $(C_n, \del_n)$ is chain homotopic to the chain complex $(C'_n, \del'_n)$ with $C'_0 = \Z$ and $C'_n = 0$ for all $n \geq 1$.