---
date: '2025-11-01'
description: covering spaces, van kampen crafted around mit 18.901 weeks 8–11.
id: fundamental group
modified: 2026-06-05 15:08:21 GMT-04:00
seealso:
  - '[[thoughts/topology/compactness|compactness and connectedness]]'
  - '[[thoughts/topology/algebraic bridge|algebraic bridge]]'
tags:
  - math
  - math/topology
title: fundamental group
created: '2025-11-01'
published: '2025-11-01'
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slug: thoughts/topology/fundamental-group
permalink: https://aarnphm.xyz/thoughts/topology/fundamental-group.md
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---
## objectives

- articulate loop concatenation, homotopy, and group structure with explicit examples.
- compute $\\pi_1$ for basic spaces (spheres, tori, projective plane, klein bottle) via van kampen.
- trace covering space classification theorems (lifting criteria, deck transformations).

## reading cues

- munkres ch. 51–56.
- mit 18.901 lectures 16–24, recitations on covering space problems.

## tasks

- [ ] derive van kampen for wedge sums and apply to bouquet of circles.
- [ ] work out covering spaces for circle, cylinder, mobius strip, torus.
- [ ] summarize relationship with group actions preparing for 18.906.