---
title: Differential Topology
description: Differential topology studies differentiable functions and manifolds in geometric spaces. Learners will understand concepts like tangent spaces, vector fields, and embedding theorems to analyze the global properties of smooth shapes.
category: mathematics
subcategory: numerical-analysis-and-topology
difficulty: beginner, intermediate, advanced
url: /subject/differential-topology
---

# Differential Topology

Differential topology studies differentiable functions and manifolds in geometric spaces. Learners will understand concepts like tangent spaces, vector fields, and embedding theorems to analyze the global properties of smooth shapes.

## Available Resources

1 Books • 1 Courses • 2 Websites

## Websites

### 1. nLab: Differential Topology

The nLab wiki entry on differential topology, a collaborative reference written by working mathematicians. It frames smooth manifolds, maps and their invariants from a higher-categorical viewpoint and links outward to related concepts, useful as a map of the field.

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

**Link:** https://ncatlab.org/nlab/show/differential+topology

**Tags:** differential-topology, smooth-manifolds, category-theory, reference

### 2. mathoverflow.net

MathOverflow is a Q&A platform for professional mathematicians to ask and answer research-level questions, offering in-depth discussions, problem-solving threads, and expert insights across advanced math topics, including differential topology.

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

**Link:** https://mathoverflow.net

**Tags:** websites, mathematics-statistics, numerical-analysis-topology

## Courses

### 1. Lectures on the Geometric Anatomy of Theoretical Physics

Explore differential topology with Frederic Schuller's "Lectures on the Geometric Anatomy of Theoretical Physics" course. Learn geometric foundations!

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

**Link:** https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic

**Tags:** differential-geometry, topology, smooth-manifolds, mathematical-physics, lie-groups

## Books

### 1. An Introduction to Differential Manifolds

**Author:** Jacques Lafontaine

A Springer graduate text that builds the theory of smooth manifolds from submanifolds of Euclidean space through tangent bundles, vector fields, Lie groups, differential forms, de Rham cohomology and integration, with many worked exercises for readers comfortable with multivariable calculus.

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

**Link:** https://www.amazon.com/dp/3319207342?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, numerical-analysis-topology

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