---
date: '2025-11-01'
description: bridge from point-set topology to algebraic topology (homology/homotopy) ahead of mit 18.906 and beyond.
id: algebraic bridge
modified: 2026-06-06 19:42:34 GMT-04:00
seealso:
  - '[[thoughts/topology/fundamental group|fundamental group]]'
  - '[[thoughts/topology/simply connected|simple connectivity]]'
  - '[[thoughts/topology/differential foundations|differential foundations]]'
  - '[[thoughts/topology/poincare|poincaré roadmap]]'
tags:
  - math
  - math/topology
title: algebraic view
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology/algebraic-bridge
permalink: https://aarnphm.xyz/thoughts/topology/algebraic-bridge.md
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full: https://aarnphm.xyz/llms-full.txt
---
## scope

{{sidenotes[phase 5-6]: this note covers phase 5 (weeks 13+) and extends into phase 6 (year 1 spring-summer) of the [[thoughts/topology/poincare|poincaré roadmap]].}}

transition from fundamental group $\pi_1$ to homology theory $H_n$. develop computational tools for topological invariants essential for:

- [[thoughts/topology/simply connected|understanding simple connectivity]] vs homological conditions
- [[thoughts/topology/3 manifolds|3-manifold topology]] (heegaard splittings, poincaré duality)
- prerequisites for [[thoughts/topology/differential foundations|differential topology]]

## fundamental group recap

from [[thoughts/topology/fundamental group|fundamental group]] phase:

- $\pi_1(X,x_0)$: homotopy classes of loops based at $x_0$
- seifert-van kampen theorem for computations
- covering space classification
- examples: $\pi_1(S^1) = \mathbb{Z}$, $\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$

_limitation_: $\pi_1$ is non-abelian, difficult to compute for complex spaces.

homology provides abelian approximation with better computational properties.

## hurewicz connection

_hurewicz theorem_: for path-connected $X$,
$H_1(X) \cong \pi_1(X)^{\text{ab}} = \pi_1(X)/[\pi_1(X),\pi_1(X)]$

abelianization of fundamental group equals first homology.

_consequence_: $\pi_1 = 0 \Rightarrow H_1 = 0$, but converse fails.

see [[thoughts/topology/simply connected#poincaré homology sphere|poincaré homology sphere]]: $H_1(\Sigma^3) = 0$ but $\pi_1(\Sigma^3) = I^*$ (order 120).

## homology theory foundations

### chain complexes

sequence of abelian groups with boundary maps:
$\cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots$

satisfying $\partial_n \circ \partial_{n+1} = 0$ (boundary of boundary is zero).

_cycles_: $Z_n = \ker(\partial_n)$ (chains with no boundary)
_boundaries_: $B_n = \text{im}(\partial_{n+1})$ (chains that are boundaries)
_homology_: $H_n = Z_n / B_n$ (cycles modulo boundaries)

### singular homology

_singular $n$-simplex_: continuous map $\sigma: \Delta^n \to X$ from standard $n$-simplex.

_singular chain group_: $C_n(X) = $ free abelian group on singular $n$-simplices.

_boundary map_: $\partial_n: C_n \to C_{n-1}$ via alternating sum of face maps.

_singular homology_: $H_n(X) = \ker(\partial_n)/\text{im}(\partial_{n+1})$.

### simplicial homology

for spaces with triangulation (simplicial complexes):

- chain groups generated by $n$-simplices in triangulation
- boundary maps via combinatorial faces
- often easier to compute than singular homology
- _theorem_: for simplicial complexes, simplicial $\cong$ singular homology

### cellular homology

for cw complexes (spaces built by attaching cells):

- chain groups: $C_n^{CW} = H_n(X^n, X^{n-1})$ (relative homology of $n$-skeleton)
- boundary maps via attaching maps
- most efficient for computation
- _theorem_: cellular $\cong$ singular homology for cw complexes

## key theorems

### homotopy invariance

_theorem_: if $f \simeq g: X \to Y$ (homotopic maps), then $f_* = g_*: H_n(X) \to H_n(Y)$.

_corollary_: homotopy equivalent spaces have isomorphic homology.

### mayer-vietoris sequence

for $X = U \cup V$ with $U,V$ open:
$\cdots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \cdots$

powerful computational tool via decomposition.

### poincaré duality

for closed oriented $n$-manifold $M^n$:
$H_k(M) \cong H^{n-k}(M)$

(homology in dimension $k$ isomorphic to cohomology in dimension $n-k$).

crucial for [[thoughts/topology/3 manifolds|3-manifold theory]].

### universal coefficient theorem

relates homology to cohomology:
$0 \to \text{Ext}(H_{n-1}(X), G) \to H^n(X;G) \to \text{Hom}(H_n(X), G) \to 0$

## computational examples

### spheres

$H_n(S^k) = \begin{cases} \mathbb{Z} & n = 0, k \\ 0 & \text{otherwise} \end{cases}$

### torus

$H_n(T^2) = \begin{cases} \mathbb{Z} & n = 0, 2 \\ \mathbb{Z}^2 & n = 1 \\ 0 & n \geq 3 \end{cases}$

via mayer-vietoris on $T^2 = D_1 \cup D_2$ (two disks along boundaries).

### projective space

$H_n(\mathbb{RP}^2) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/2\mathbb{Z} & n = 1 \\ 0 & n \geq 2 \end{cases}$

### poincaré homology sphere

$H_n(\Sigma^3) = \begin{cases} \mathbb{Z} & n = 0, 3 \\ 0 & n = 1, 2 \end{cases}$

same as $S^3$, but $\pi_1(\Sigma^3) = I^* \neq \{e\}$.

## cw complex examples

### sphere $S^n$

- one 0-cell (point)
- one $n$-cell attached via constant map

### torus $T^2$

- one 0-cell
- two 1-cells (meridian, longitude)
- one 2-cell attached via $aba^{-1}b^{-1}$

### projective plane $\mathbb{RP}^2$

- one 0-cell
- one 1-cell (loop)
- one 2-cell attached via $aa$ (double covering)

### klein bottle

- one 0-cell
- two 1-cells
- one 2-cell attached via $aba^{-1}b$

## tasks for mit 18.906 preparation

- [x] understand chain complexes and exact sequences
- [x] compute $H_*(S^n)$ using cellular homology
- [ ] work through mayer-vietoris for torus computation
- [ ] verify poincaré homology sphere has $H_1 = 0$
- [ ] construct cw complex structures for standard spaces
- [ ] practice long exact sequences of pairs
- [ ] understand universal coefficient theorem statement

## higher homotopy groups (preview)

beyond $\pi_1$ and homology, there are higher homotopy groups $\pi_n(X)$:
$\pi_n(X,x_0) = [(S^n,s_0), (X,x_0)]$

homotopy classes of based maps from $n$-sphere.

_properties_:

- $\pi_n$ is abelian for $n \geq 2$
- hurewicz connects to homology: $H_n(X) = \pi_n(X)$ for simply connected spaces (first non-trivial)
- very difficult to compute in general (e.g., $\pi_3(S^2) = \mathbb{Z}$ via hopf fibration)

not needed for poincaré conjecture, but important for higher topology.

## connection to poincaré roadmap

this phase (5-6) completes year 1 foundations:

- weeks 1-12: point-set + fundamental group ([[thoughts/topology|main hub]])
- weeks 13+: homology theory (this note)
- spring semester: mit 18.906 covering hatcher ch 2-3

_next steps_:

- year 2: [[thoughts/topology/differential foundations|differential topology]] and [[thoughts/topology/3 manifolds|3-manifold theory]]
- understand why $\pi_1=0$ is the right condition for poincaré ([[thoughts/topology/simply connected|simple connectivity]])

see [[thoughts/topology/poincare|full roadmap]] for integration into 4-year plan.

## resources

primary:

- hatcher “algebraic topology” ch 2 (homology)
- munkres “elements of algebraic topology”

## exercises

1. compute $H_*(S^1 \vee S^1)$ (wedge of two circles) using mayer-vietoris
2. show $H_*(\mathbb{RP}^n)$ has $\mathbb{Z}/2\mathbb{Z}$ in odd dimensions up to $n$
3. verify hurewicz: $H_1(T^2) = \mathbb{Z}^2 = \pi_1(T^2)^{\text{ab}}$
4. compute cellular chain complex for $\mathbb{CP}^2$
5. use poincaré duality: $H_1(M^3) \cong H^2(M^3)$ for closed oriented 3-manifold
6. prove $H_n(X \times Y) = \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y)$ (künneth)
7. compute homology of lens space $L(5,2)$ using covering space $S^3 \to L(5,2)$