---
date: '2025-11-01'
description: structures, decompositions, and geometric properties of 3-dimensional manifolds.
id: 3 manifolds
modified: 2026-06-05 15:08:21 GMT-04:00
tags:
  - math
  - math/topology
  - 3-manifolds
title: 3-manifold topology
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology/3-manifolds
permalink: https://aarnphm.xyz/thoughts/topology/3-manifolds.md
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full: https://aarnphm.xyz/llms-full.txt
---
## scope

{{sidenotes[dimension 3]: special—enough structure for rich theory, yet tractable. the poincaré conjecture and geometrization are fundamentally 3-dimensional phenomena.}}

comprehensive study of closed 3-manifolds. understand why dimension 3 is goldilocks: not too rigid (like dim 2), not too flexible (like dim 4+).

this phase bridges [[thoughts/topology/algebraic bridge|algebraic topology]] and [[thoughts/topology/ricci flow|ricci flow]] by providing geometric-topological foundations.

## major topics

### examples and constructions

- $S^3$: unit quaternions, hopf fibration $S^1 \to S^3 \to S^2$
- lens spaces $L(p,q)$: quotients of $S^3$ by cyclic group action
- [[thoughts/topology/simply connected#poincaré homology sphere|poincaré homology sphere]]: $(5,2,2)$ surgery on trefoil
- torus bundles: $T^2$ bundles over $S^1$ with monodromy
- seifert fibered spaces: circle bundles over surfaces

### surgery theory

- dehn surgery on knots: remove $N(K) \cong S^1 \times D^2$, reglue
- $(p,q)$ surgery: meridian to $p \cdot \text{longitude} + q \cdot \text{meridian}$
- lickorish-wallace theorem: every closed 3-manifold obtained by surgery on link in $S^3$
- kirby calculus: relating different surgery presentations

### heegaard decompositions

- handlebody: solid genus-$g$ object (tubular neighborhood of graph)
- heegaard splitting: $M = H_1 \cup_\phi H_2$ glued along boundary
- heegaard genus: minimal genus of splitting
- heegaard diagrams: encoding via curves on surface

### decomposition theorems

prime decomposition (kneser-milnor):

- every closed 3-manifold: $M = P_1 \# P_2 \# \cdots \# P_k$ (unique up to order)
- prime: can’t be written as nontrivial connected sum
- $S^3$ is prime (simplest)

jsj decomposition:

- canonical decomposition along incompressible tori
- decomposes into seifert pieces and hyperbolic pieces
- atoroidal manifolds: no essential tori

### thurston’s eight geometries

every closed 3-manifold admits geometric decomposition into pieces, each with one of:

1. spherical $S^3$: constant positive curvature
2. euclidean $E^3$: flat
3. hyperbolic $H^3$: constant negative curvature
4. $S^2 \times \mathbb{R}$: product geometry
5. $H^2 \times \mathbb{R}$: product geometry
6. $\widetilde{SL_2\mathbb{R}}$: universal cover of unit tangent bundle of hyperbolic plane
7. nil: heisenberg group
8. sol: solvable but not nilpotent lie group

see also: {{sidenotes[geometrization]: perelman proved thurston’s geometrization conjecture, which implies poincaré. simply connected manifolds must be spherical.}}

## why 3-manifolds are special

dimension 2: complete classification (sphere, torus, higher genus surfaces, non-orientable)

- simple, well-understood
- gauss-bonnet relates curvature to topology

dimension 3: richest theory

- poincaré conjecture, geometrization
- fundamental group central role
- knot theory intertwined

dimension 4+: wild west

- exotic structures on $\mathbb{R}^4$
- poincaré fails (counterexamples exist for dim 4+)
- h-cobordism theorem works (dim $\geq 5$)

## connection to poincaré conjecture

[[thoughts/topology/simply connected|simply connected]] 3-manifolds can’t have non-trivial prime or jsj decomposition:

- connected sum with $\pi_1 \neq 0$ factor breaks simple connectivity
- incompressible torus in simply connected manifold bounds ball on one side

forces single geometric piece. simply connected + compact + geometric structure $\Rightarrow$ must be $S^3/\Gamma$ with $\Gamma \subset SO(4)$ finite. simple connectivity forces $\Gamma = \{e\}$, hence $M \cong S^3$.

the [[thoughts/topology/ricci flow|ricci flow]] proof by perelman establishes geometrization.

## computational tools

### snappy (python)

```python
import snappy

M = snappy.Manifold('m004')  # figure-eight knot complement
print(M.volume())  # hyperbolic volume
print(M.homology())  # homology groups
```

hyperbolic structures, dehn surgery, geodesics.

### regina

triangulations, normal surfaces, 0-efficiency, fundamental group presentations.

## exercises placeholder

1. construct $\mathbb{RP}^3$ as $S^3/(\mathbb{Z}/2\mathbb{Z})$ quotient
2. compute $\pi_1(L(5,1))$ from heegaard diagram
3. show $(1,0)$ surgery on unknot gives $S^1 \times S^2$
4. verify $S^3 = H_1 \cup_\phi H_1$ (genus 1 heegaard splitting)
5. compute homology of poincaré homology sphere via mayer-vietoris
6. show $T^3$ has euclidean geometry
7. find heegaard diagram for trefoil knot complement
8. prove $\mathbb{RP}^3 \# \mathbb{RP}^3$ is not prime

(full problem sets to be added during year 2-3 study)

## resources

primary texts:

- hempel “3-manifolds” (foundational)
- thurston “three-dimensional geometry and topology” (geometric perspective)
- rolfsen “knots and links” (surgery constructions)

see [[thoughts/topology/resources#phase 8: 3-manifold topology (year 2-3, parallel)|3-manifold resources]] for complete list.

## timeline

year 2-3 (parallel with [[thoughts/topology/differential foundations|differential topology]] and [[thoughts/manifold|riemannian geometry]]):

- fall year 2: basic constructions (hempel ch 1-4)
- spring year 2: heegaard theory, jsj (hempel ch 5-8)
- year 3: geometric structures (thurston ch 1-3)

## further reading

- [[thoughts/topology/ricci flow|ricci flow]] (evolution on 3-manifolds)
- [[thoughts/topology/poincare|poincaré proof]] (geometrization via ricci flow)