---
title: Morse Theory
description: Morse theory analyzes the topology of a manifold by studying differentiable functions on it and their critical points. Learners will understand how to reconstruct manifolds using cell complexes and relate critical points to global topological invariants.
category: mathematics
subcategory: differential-topology
difficulty: beginner, intermediate, advanced
url: /subject/morse-theory
---

# Morse Theory

Morse theory analyzes the topology of a manifold by studying differentiable functions on it and their critical points. Learners will understand how to reconstruct manifolds using cell complexes and relate critical points to global topological invariants.

## Available Resources

1 Books • 3 Websites

## Websites

### 1. nLab: Morse Theory

The nLab wiki entry summarizing how critical points of a smooth function determine a manifold's topology, including Morse functions, handle decompositions, Morse homology and the Morse inequalities, with a curated bibliography. Serves as a terse reference and reading map.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/Morse+theory

**Tags:** morse-theory, differential-topology, morse-homology, smooth-manifolds

### 2. 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

### 3. 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

## Books

### 1. Morse Theory

**Author:** John Milnor

John Milnor's 1963 Princeton lecture notes, the classic account of how nondegenerate critical points of a function build a manifold cell by cell. Continues into geodesics and the path-space calculus of variations, ending with a proof of Bott periodicity.

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

**Link:** https://www.amazon.com/dp/0691080089?tag=edmonddante07-20

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

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