---
title: Smooth Maps
description: Smooth maps are infinitely differentiable functions between manifolds. Learners will understand how to analyze critical points, apply Sard's theorem, and use transversality to study the intersections and preimages of submanifolds.
category: mathematics
subcategory: differential-topology
difficulty: beginner, intermediate, advanced
url: /subject/smooth-maps
---

# Smooth Maps

Smooth maps are infinitely differentiable functions between manifolds. Learners will understand how to analyze critical points, apply Sard's theorem, and use transversality to study the intersections and preimages of submanifolds.

## Where to start

Start with Lectures on Differential Topology by Alexander Kupers, free lecture notes with complete proofs that follow the same transversality-first order as the standard course text. If you only use one resource, make it Differential Topology by Victor Guillemin and Alan Pollack, a paid textbook built around transversality that works through preimages of regular values and transverse intersections with exercises.

## Available Resources

1 Books • 3 Websites

## Books

### 1. Differential Topology (Graduate Texts in Mathematics 33)

**Author:** Morris W. Hirsch

The reference for the topology on spaces of smooth maps and for transversality theorems in their general (jet, parametric) forms. Graduate text by Morris Hirsch treating spaces of smooth maps with the weak and strong Whitney topologies, approximation theorems, the Morse-Sard theorem, and jet and parametric transversality. Readers can prove genericity statements about maps, embeddings and intersections in full generality.

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

**Link:** https://www.amazon.com/dp/0387901485?tag=edmonddante07-20

### 2. Differential Topology

**Author:** Victor Guillemin, Alan Pollack

Of the standard texts this is the one built explicitly around transversality, half of the stated learning goal; it works through preimages of regular values and transverse intersections in full detail with exercises where Milnor is terse. The standard course text, organized around transversality as its unifying idea. Covers smooth maps, the stack-of-records and preimage theorems, measure zero and Sard's theorem, oriented intersection theory, Lefschetz fixed points and Poincare-Hopf.

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

**Link:** https://www.amazon.com/dp/0821851934?tag=edmonddante07-20

**Tags:** differential-topology, smooth-maps, transversality, sard's-theorem, intersection-theory, smooth-manifolds

## Websites

### 1. Lectures on Differential Topology

**Author:** Alexander Kupers

The best free full-course substitute for Guillemin and Pollack: complete proofs, lecture-by-lecture structure, and the same transversality-first organization, so a learner without book access loses nothing on the core material.  Free 300-page lecture notes from Harvard Math 132 and Toronto MAT1300. Treats smooth maps and derivatives, immersions, submersions, embeddings, the Whitney embedding theorem, transversality and the preimage theorem, then intersection theory and Morse theory.

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

**Link:** https://www.utsc.utoronto.ca/people/kupers/teaching/differential-topology-lecture-notes/

**Tags:** differential-topology, smooth-maps, transversality, whitney-embedding-theorem, morse-theory, lecture-notes

### 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

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