---
title: 'lecture 02: homology'
description: semi-simplicial sets, reversing a 1-simplex up to boundary, homology of the empty space and a point, chain maps and induced maps on homology
date: '2026-10-08'
modified: 2026-10-09 09:01:29 GMT-04:00
tags:
  - course
  - mit
  - math/topology
created: '2026-10-08'
published: '2026-10-08'
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slug: courses/18.905-fall-2016/lectures/02
permalink: https://aarnphm.xyz/courses/18.905-fall-2016/lectures/02.md
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---
Source: [[courses/18.905-fall-2016/resources/mit18_905f16_lec2|lecture 2]]. The lecture packages last time’s construction as a pipeline, computes the homology of the empty space and a point, and proves that continuous maps induce maps on homology.

## the pipeline

For $n \geq 1$ and $0 \leq i \leq n$, the face maps $d_i : \mathrm{Sin}_n(X) \to \mathrm{Sin}_{n-1}(X)$ satisfy the quadratic relations from the homework. A collection of sets $K_n$ ($n \geq 0$) with face maps $d_i : K_n \to K_{n-1}$ in positive degrees satisfying those relations is a semi-simplicial set. So

$$
\{\text{spaces}\} \xrightarrow{\ \mathrm{Sin}_*\ } \{\text{semi-simplicial sets}\} \xrightarrow{\ \mathbb{Z}(-)\ } \{\text{semi-simplicial abelian groups}\} \longrightarrow \{\text{chain complexes}\} \xrightarrow{\ H_*\ } \{\text{graded abelian groups}\}.
$$

The third arrow forms $d = \sum_{i=0}^n (-1)^i d_i$ in positive degrees and sets the boundary out of degree zero to zero.

## reversing a 1-simplex

Take $\sigma : \Delta^1 \to X$ and let $\bar\sigma = \sigma \circ \phi$ where $\phi(t, 1-t) = (1-t, t)$ flips the interval. The reversed map is a generator of $S_1(X)$, while $-\sigma$ is a chain with coefficient $-1$, so $\bar\sigma \neq -\sigma$. Their sum is a boundary, giving $\bar\sigma \equiv -\sigma \bmod B_1(X)$. If $\sigma$ is a cycle, this gives $\lbrack \bar\sigma \rbrack = -\lbrack \sigma \rbrack$ in $H_1(X)$.

The 2-chain: let $\pi : \Delta^2 \to \Delta^1$ be the affine map with $e_0, e_2 \mapsto e_0$ and $e_1 \mapsto e_1$. Put $x = \sigma(e_0)$ and write $c^n_x$ for the constant $n$-simplex at $x$. Then

$$
d(\sigma \circ \pi) = \sigma\pi d^0 - \sigma\pi d^1 + \sigma\pi d^2 = \bar\sigma - c^1_x + \sigma.
$$

The constant edge is itself a boundary: $d\, c^2_x = c^1_x - c^1_x + c^1_x = c^1_x$. Adding this constant 2-simplex cancels the negative edge term:

$$
\sigma + \bar\sigma = d\big(\sigma \circ \pi + c^2_x\big).
$$

Language: two cycles differing by a boundary $dc$ are homologous, and $c$ is a homology between them.

## two computations

**Empty space.** $\mathrm{Sin}_n(\emptyset) = \emptyset$, so $S_*(\emptyset) = 0$ and $H_*(\emptyset) = 0$.

**A point.** $\mathrm{Sin}_n(*) = \{c^n_*\}$, so $S_n(*) = \mathbb{Z}$ for $n \geq 0$. For $n \geq 1$, every face of $c^n_*$ is $c^{n-1}_*$. The alternating signs therefore give

$$
d(c^n_*) = \sum_{i=0}^n (-1)^i c^{n-1}_* =
\begin{cases}
c^{n-1}_* & n \geq 2 \text{ even} \\
0 & n \geq 1 \text{ odd.}
\end{cases}
$$

Degree zero is separate: $d(c^0_*) = 0$, since $S_{-1}(*) = 0$. The complex is $0 \leftarrow \mathbb{Z} \xleftarrow{0} \mathbb{Z} \xleftarrow{1} \mathbb{Z} \xleftarrow{0} \mathbb{Z} \xleftarrow{1} \cdots$. Cycles and boundaries agree except in degree zero, where $B_0 = 0$ and $Z_0 = \mathbb{Z}$. So $H_0(*) = \mathbb{Z}$ and $H_i(*) = 0$ for $i \neq 0$.

## induced maps

A continuous $f : X \to Y$ gives $f_* : \mathrm{Sin}_n(X) \to \mathrm{Sin}_n(Y)$, $\sigma \mapsto f \circ \sigma$. It commutes with faces: $d_i f_* \sigma = f \circ \sigma \circ d^i = f_* d_i \sigma$. A diagram is commutative when all composites with the same source and target agree. Passing to free abelian groups keeps the square commutative.

A chain map $f : C_* \to D_*$ is a family of group homomorphisms $f_n : C_n \to D_n$ with $d_D f_n = f_{n-1} d_C$. So $f_* : S_*(X) \to S_*(Y)$ is a chain map.

**A chain map induces $f_* : H_n(C) \to H_n(D)$.** The first diagram chase of the course:

- cycles go to cycles: if $d_C c = 0$ then $d_D f_n(c) = f_{n-1} d_C c = 0$;
- boundaries go to boundaries: if $c = d_C c'$ then $f_n(c) = f_n d_C c' = d_D f_{n+1}(c')$.

The induced map sends $[c]$ to $[f_n(c)]$. Changing $c$ by a boundary changes $f_n(c)$ by a boundary, so the result is independent of the chosen representative.

The lecture ends by opening lecture 3: categories, functors, natural transformations, which recast “spaces to graded abelian groups” as a functor.

## problems

1. Verify the sign in $d(\sigma \circ \pi)$ by listing the three faces of $\sigma \circ \pi$ as maps $\Delta^1 \to X$.
2. Show directly that a constant 1-simplex is a boundary. (It is $d$ of the constant 2-simplex.)
3. Compute $H_*$ of a two-point space by the same method as for a point.