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Rational Homotopy Theory and Differential Forms

Authors: Phillip Griffiths, John Morgan

Publisher: Springer New York

Book Series : Progress in Mathematics

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About this book

This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented.

New to the Second Edition:

*Fully-revised appendices including an expanded discussion of the Hirsch lemma

*Presentation of a natural proof of a Serre spectral sequence result

*Updated content throughout the book, reflecting advances in the area of homotopy theory

With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

This chapter is an informal introduction to the material in the rest of this book. The goal of the book is to relate smooth differential forms of a manifold to classical algebra-topological invariants. The next seven chapters are a review of the basics of classical homotopy theory including the Postnikov tower of a space and the localization of a homotopy type at 0, forming its rational homotopy type. Chapters 9–15 introduce differential forms and their homotopy theory of differential graded algebras and relate these notions to those of rational homotopy theory as introduced in Chaps. 2–8.

Phillip Griffiths, John Morgan

Chapter 2. Basic Concepts

Abstract

This chapter begins with a brief review of CW complexes and basic notions of homotopy theory. Fibrations are introduced and the homotopy exact sequence of a fibration is proved. The path space and loop space are introduced. Singular homology and cohomology are defined, and their basic properties are enumerated. Products in homology and cohomology are introduced and Poincar duality is stated. The chapter ends with a brief introduction to the functorial language.

Phillip Griffiths, John Morgan

Chapter 3. CW Homology Theorem

Abstract

This chapter states and proves the main result about the singular homology of a CW complex, the CW homology theorem. This result states that the homology of a CW complex can be computed from a chain complex whose chain groups have a generator for each cell and whose boundary map is determined from topological information about the attaching maps of the cells. The chapter finishes with some simple examples showing how to use the main result to make computations.

Phillip Griffiths, John Morgan

Chapter 4. The Whitehead Theorem and the Hurewicz Theorem

Abstract

Chapter 4 introduces the homotopy groups of a space with a base point and establishes several basic results about these groups. The Hurewicz homomorphism from these groups to the homology groups is defined. Whitehead’s theorem that a map between CW complexes inducing an isomorphism on homotopy groups is a homotopy equivalence is stated and proved. Brouwer’s theorem computing the \(\mathrm{n}\) th homotopy group of the n-sphere is stated and proved. The Hurewicz theorem, which states that for a simply connected space the first nonzero homotopy group and reduced homology group are isomorphic, is proved. Lastly, the homotopy exact sequence of a fibration is stated and proved.

Phillip Griffiths, John Morgan

Chapter 5. Spectral Sequence of a Fibration

Abstract

This chapter is devoted to the Leray–Serre spectral sequence for the homology and cohomology of a fibration. Spectral sequences are introduced in the general context of filtered chain and cochain complexes. The Leray–Serre spectral sequence for the cohomology and the homology of a fibration is introduced and the E2 term is evaluated. Examples such as the cohomology of complex projective space and the Grassmannians and the rational cohomology of the Eilenberg–MacLane spaces are computed.

Phillip Griffiths, John Morgan

Chapter 6. Obstruction Theory

Abstract

Recall, in the proof of the Whitehead theorem, we showed that if X is a CW complex and Y is a space with \(\pi _{i}(\mathrm{Y}) = 0\) for all i ≥ 0, then any map \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}\) is homotopic to the base point map \(\mathrm{X} \rightarrow \mathrm{ y}_{0} \in \mathrm{ Y}\). The proof was by induction over the skeleta of X. Obstruction theory is a generalization of this technique to the case when the homotopy groups of Y are not necessarily zero. It does not give a complete understanding of when a map \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}\) is homotopic to a constant, or more generally when two maps \(\mathrm{f}_{1},\mathrm{f}_{2}: \mathrm{X} \rightarrow \mathrm{ Y}\) are homotopic, but it gives some insight.

Phillip Griffiths, John Morgan

Chapter 7. Eilenberg–MacLane Spaces, Cohomology, and Principal Fibrations

Abstract

This chapter begins by showing that maps of a CW complex to an Eilenberg–MacLane space are classified by the elements in a cohomology group. Then principal fibrations with fiber an Eilenberg–MacLane space are classified by elements in a similar cohomology group.

Phillip Griffiths, John Morgan

Chapter 8. Postnikov Towers and Rational Homotopy Theory

Abstract

This chapter begins with the inductive construction (indexed by n) of the nth-stage of the Postnikov tower of a simply connected space. Next, the notion of localizing an abelian group at 0 is introduced, and the definition of a Q-space is given and then the existence and uniqueness of the localization of a space at 0 is established by localizing the Postnikov tower.

Phillip Griffiths, John Morgan

Chapter 9. deRham’s Theorem for Simplicial Complexes

Abstract

This chapter begins with a definition of the piecewise linear rational polynomial forms on a simplicial complex. These are shown to compute the usual rational singular cohomology (the p.l. deRham theorem). The construction is shown to be natural under subdivision of the simplicial complex and to produce a multiplicative isomorphism to singular cohomology. Connections with the usual smooth deRham theorem are given using piecewise smooth forms on a smoothly triangulated manifold.

Phillip Griffiths, John Morgan

Chapter 10. Differential Graded Algebras

Abstract

This chapter introduces (commutative) differential graded algebras (DGAs). The notion of a minimal DGA is introduced. These are constructed using Hirsch extensions. Hirsch extensions, up to isomorphism, are classified by a cohomology group. Lastly, the minimal model of a general simply connected DGA is defined and proved to exist by an inductive construction whose inductive step is a Hirsch extension.

Phillip Griffiths, John Morgan

Chapter 11. Homotopy Theory of DGAs

Abstract

The chapter begins by introducing the notion of a homotopy of DGA maps. Then, an obstruction theory lifting maps and homotopies over a Hirsch extension is established. This obstruction theory is used to show that homotopy is an equivalence relation on maps from a given minimal DGA to another given DGA. From this, one establishes that any two minimal models for a given DGA are isomorphic by an isomorphism making the relevant diagram commute up to homotopy. Also, one shows that a map between minimal DGAs inducing an isomorphism on cohomology is an isomorphism.

Phillip Griffiths, John Morgan

Chapter 12. DGAs and Rational Homotopy Theory

Abstract

This chapter connects the two themes of the book—rational homotopy theory and differential forms. There is an equivalence between Hirsch extensions of the algebra of p.l. forms on a simplicial complex and principal fibrations over the space with fiber an Eilenberg–MacLane space. This equivalence is seen by identifying the cohomology groups that classify each. The basic theorem is that under this correspondence, the Hirsch extension maps to the p.l. forms on the total space of the principal fibration, extending the map on the base and inducing an isomorphism on cohomology. Once this result is established, inductively one shows that the rational Postnikov tower of a space is read off from the minimal model of the p.l. forms on the space. The proof of the main inductive result, the Hirsch lemma, is postponed until Chap. 16.

Phillip Griffiths, John Morgan

Chapter 13. The Fundamental Group

Abstract

Until now we have been working with simply connected spaces, but this chapter takes up the question of the fundamental group and its relation to differential forms. The notion of a 1-minimal model is introduced, and it is shown that every connected DGA has a 1-minimal model, unique up to isomorphism. The dual of this is a tower of nilpotent Lie algebras. We then turn to the nilpotent quotients of the fundamental group. The main result is that the tower of nilpotent rational Lie algebras associated with tower of nilpotent Lie group quotients of the fundamental group of a space is dual to the 1-minimal model of the p.l. forms on the space.

Phillip Griffiths, John Morgan

Chapter 14. Examples and Computations

Abstract

This chapter considers various examples and makes explicit computations. These include the spheres, complex projective spaces, the Borromean rings, symmetric spaces, and the classifying spaces BU(n) and BU. We also consider the graded Lie algebra defined by Whitehead products on homotopy groups. We consider the third homotopy group of a simply connected space and the homotopy theory of simply connected 4-manifolds. Lastly, we discuss products and Massey products in cohomology, the principle of two types and the formality of compact Kahler manifolds.

Phillip Griffiths, John Morgan

Chapter 15. Functorality

Abstract

We show that the assignment to a DGA map, the induced map, defined up to homotopy, between the minimal models is functorial. We show that for a local space C and for any simplicial complex X, the map from homotopy classes of maps from X to C to the homotopy classes of maps from the minimal model of the p.l. forms on C to that for X is a functorial bijection. We reformulate this result as an equivalence of the rational homotopy category of simply connected spaces with the homotopy category of simply connected DGAs defined over the rationals.

Phillip Griffiths, John Morgan

Chapter 16. The Hirsch Lemma

Abstract

In this chapter, we prove the main technical result from Chap. 12, namely, the result comparing principle bundles and Hirsch extensions.

Phillip Griffiths, John Morgan

Chapter 17. Quillen’s Work on Rational Homotopy Theory

Abstract

In this chapter we review Quillen’s work on rational homotopy theory. Quillen gives a sequence of rational homotopy categories and proves that they are all equivalent. To a simply connected space, he associates a simplicial set with trivial one-skeleton and then the simplicial loop group. From there he passes to simplicial (complete) Hopf algebra, then to a simplicial Lie algebra, to a differential graded Lie algebra, and finally to a differential graded co-algebra. The latter dualizes to a differential graded algebra homotopy equivalent to the p.l. forms on the space, so that Quillen’s construction agrees in homotopy theory with Sullivan’s.

Phillip Griffiths, John Morgan

Chapter 18. A ∞ -Structures and C ∞ -Structures

Abstract

In this chapter we review the notions of A-infinity algebras and A-infinity categories and discuss briefly their appearance in symplectic topology. We then state the result that homotopy theory of commutative DGAs is equivalent to the homotopy theory of commutative A-infinity algebras, the so-called C-infinity algebras. It follows that there is a C-infinity map from the cohomology of a space to its minimal model, a map which induces the identity on cohomology.

Phillip Griffiths, John Morgan

Chapter 19. Exercises

Abstract

The last chapter gives a series of exercises related to the material in the rest of the book.

Phillip Griffiths, John Morgan

Backmatter

Title: Rational Homotopy Theory and Differential Forms
Authors: Phillip Griffiths
John Morgan
Copyright Year: 2013
Publisher: Springer New York
Electronic ISBN: 978-1-4614-8468-4
Print ISBN: 978-1-4614-8467-7
DOI: https://doi.org/10.1007/978-1-4614-8468-4

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