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# 55P: Homotopy theory

## Introduction

This is the study of topological spaces using a broader equivalence than homeomorphism. It's really the natural setting for algebraic topology because the traditional algebraic invariants (fundamental group, etc.) are isomorphic not only for homeomorphic spaces but for spaces of the same homotopy type. Since most of the information about these invariants is already on the algebraic topology page, there isn't much left here. There are some comments about contractibility.

## History

Rather narrow for our purposes but a place to start: Moore, J. C.; Neisendorfer, J. A.: "A view of some aspects of unstable homotopy theory since 1950", in Homotopy theory -- Durham 1985, 117--148, London Math. Soc. Lecture Note Ser., 117, Cambridge Univ. Press, Cambridge-New York, 1987. MR89e:55002

## Applications and related fields

For simple homotopy type, see 57Q10

## Subfields

• 55P05: Homotopy extension properties, cofibrations
• 55P10: Homotopy equivalences
• 55P15: Classification of homotopy type
• 55P20: Eilenberg-Mac Lane spaces
• 55P30: Eckmann-Hilton duality
• 55P35: Loop spaces
• 55P40: Suspensions
• 55P42: Stable homotopy theory, spectra
• 55P43: Spectra with additional structure (E_\infty, A_\infty, ring spectra, etc.) [new in 2000]
• 55P45: H-spaces and duals
• 55P47: Infinite loop spaces
• 55P57: Proper homotopy theory [new in 2000]
• 55P60: Localization and completion
• 55P62: Rational homotopy theory
• 55P65: Homotopy functors
• 55P92: Relations between equivariant and nonequivariant homotopy theory [new in 2000]
• 55P99: None of the above but in this section

Parent field: 55: Algebraic Topology

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Recent texts have tended to be more specialized. Appropriate for all of section 55P is the book of Whitehead, George W.: "Elements of homotopy theory", GTM 61, Springer-Verlag, New York-Berlin, 1978. 744 pp. ISBN 0-387-90336-4 MR80b:55001