---
title: Fundamental Group
description: The fundamental group is an algebraic invariant that measures the loops and 1-dimensional holes within a topological space. Learners will understand how to compute this group using the Seifert-van Kampen theorem and classify covering spaces.
category: mathematics
subcategory: algebraic-topology
difficulty: beginner, intermediate, advanced
url: /subject/fundamental-group
---

# Fundamental Group

The fundamental group is an algebraic invariant that measures the loops and 1-dimensional holes within a topological space. Learners will understand how to compute this group using the Seifert-van Kampen theorem and classify covering spaces.

## Available Resources

1 Books • 1 Courses • 4 Websites

## Websites

### 1. nLab: Fundamental Group

The nLab's entry on the fundamental group, treating it from a category-theoretic viewpoint: as an automorphism group in the fundamental groupoid, with links to covering spaces, homotopy groups and higher generalizations. Best read after a standard course in algebraic topology.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/fundamental+group

**Tags:** fundamental-group, algebraic-topology, homotopy-theory, category-theory

### 2. Hatcher's Algebraic Topology (free PDF)

**Author:** Allen Hatcher

Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

**Tags:** algebraic-topology, cw-complexes, homology, homotopy-theory, fundamental-group

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 4. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Courses

### 1. Introduction to Algebraic Topology

Learn algebraic topology with N J Wildberger! This course explores the fundamental group using a novel approach.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PL_bVGic_CrGs5bOLW_RR-8Y2RM09Fyg_0

**Tags:** algebraic-topology, fundamental-group, homotopy, surfaces

## Books

### 1. Topology

**Author:** James R. Munkres

James Munkres's standard textbook on topology, in two parts: general point-set topology (spaces, continuity, connectedness, compactness, countability and separation axioms) and introductory algebraic topology, including the fundamental group, covering spaces and the Seifert-van Kampen theorem. Proof-based; suits readers comfortable with real analysis.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/9332549532?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, algebraic-topology

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