Skip to main content
Dantes.io Logo
Dantes.io

Best resources to learnSmooth Maps

Study sheet

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.

5 resources · · How we curate

📚Books(2)

bookPaid

Differential Topology (Graduate Texts in Mathematics 33)

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.

advanced🇬🇧
bookPaid

Differential Topology

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.

intermediate🇬🇧

🌐Websites(3)

website

Lectures on Differential Topology

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.

advanced🇬🇧
website

Wolfram MathWorld

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.

intermediate🇬🇧
website

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.

advanced🇬🇧
Share this topic

More in Differential Topology

See all 10 topics in Differential Topology →

Frequently Asked Questions

Where should I start learning Smooth Maps?

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.

What are the best free resources to learn Smooth Maps?

Dantes has curated 4 resources for Smooth Maps, including 1 books, 3 websites. All resources are hand-picked for quality — no algorithmic filler. Browse the full list above to find the format that works best for you.

Is Smooth Maps hard to learn?

Smooth Maps is approachable at the beginner level — there are resources here specifically for those starting from scratch. As you progress, intermediate and advanced material is also available to take your skills further.

What types of Smooth Maps learning resources are available on Dantes?

For Smooth Maps, Dantes has curated 1 books, 3 websites. Each resource type serves a different learning style: videos and YouTube for visual learners, books for depth, courses for structured progression, and websites for quick reference.

How does Dantes select Smooth Maps resources?

Dantes is an algorithm-free learning directory. Resources are hand-curated based on quality, accuracy, and usefulness — not engagement metrics or paid placements. The goal is to surface the best learning material for Smooth Maps, whether it's a free YouTube series, a classic textbook, or an open courseware from a top university.

Go deeper

Test your understanding of Smooth Maps

Explain it out loud. An AI tutor listens and asks questions that expose gaps you didn't know you had.