Rational Homotopy Theory and Differential Forms - Revised Edition

In this review of Rational Homotopy Theory and Differential Forms, the reviewer finds a rigorous, modern exposition aimed at graduate students and researchers working in algebraic topology. The single biggest reason to buy this book is its careful connection between homotopy-theoretic ideas and differential forms, with the second edition providing updated appendices and a clarified proof of a Serre spectral sequence result. Readers looking for a focused, technical treatment of rational homotopy theory will appreciate the clarity and the inclusion of Sullivan's approach to computing rational homotopy type from forms.

Key Features

• Revised appendices: The expanded discussion of the Hirsch lemma offers clearer background and makes several technical arguments more accessible for study and reference.
• Postnikov and rational towers: Detailed treatment of Postnikov towers helps readers understand how rationalization interacts with classical homotopy decompositions.
• Differential forms focus: The book links differential forms to topology through de Rham theorems on simplicial complexes, strengthening computational intuition.
• Sullivan's results presented: A presentation of Sullivan's methods shows how to compute rational homotopy types directly from differential forms, useful for hands-on applications.
• Corrected and updated content: The second edition updates material throughout to reflect advances in homotopy theory, improving accuracy and topical relevance.

Who It's For

This book is best suited to graduate students in mathematics, and researchers in algebraic topology or differential geometry who already have some background in homotopy theory and differential forms. Its emphasis on modern approaches and updated appendices makes it a useful textbook for a reading course or a reference for active researchers.

Those without prior exposure to basic algebraic topology or de Rham theory should look elsewhere for a more introductory treatment, since the text assumes familiarity with core topological and algebraic techniques rather than developing them from first principles.

Pros & Cons

Pros

• Concise presentation of Postnikov towers and rationalization that clarifies conceptual structure.
• Careful link between differential forms and topology via de Rham-type results on simplicial complexes.
• Updated appendices and corrected proofs increase reliability as a reference for researchers.
• Includes Sullivan's computational methods, valuable for explicit computations in rational homotopy.

Cons

• Dense, technical style assumes background knowledge and is not aimed at beginners.
• Focused scope means readers seeking broad introductions to algebraic topology may need supplemental texts.

Specifications

Title	Rational Homotopy Theory and Differential Forms
Authors	Phillip Griffiths, John Morgan
Edition	Second edition, revised and corrected
Series	Progress in Mathematics, 16
Primary topics	Rational homotopy theory; differential forms; Postnikov towers
New material	Expanded appendices, Hirsch lemma discussion, updated proofs

Our Verdict

Rational Homotopy Theory and Differential Forms is a strong, well-updated reference for graduate students and researchers who need a focused bridge between homotopy theory and differential forms. Its corrected proofs and expanded appendices make it good value for those seeking rigorous, usable presentations of Sullivan's methods and de Rham results on simplicial complexes.

Frequently Asked Questions

Does this edition include new corrections?
Yes, the second edition is revised and corrected with expanded appendices and clarified proofs, including a fuller discussion of the Hirsch lemma.

Is prior topology required?
Yes, readers should have background in basic algebraic topology and differential forms to follow the material comfortably.

Does the book cover computational methods?
It presents Sullivan's approach to computing rational homotopy type from forms, making it useful for explicit computations.