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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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
• 55P25: Spanier-Whitehead duality
• 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
• 55P48: Loop space machines, operads [See also 18D50] [new in 2000]
• 55P55: Shape theory, See also 54C56, 55Q07
• 55P57: Proper homotopy theory [new in 2000]
• 55P60: Localization and completion
• 55P62: Rational homotopy theory
• 55P65: Homotopy functors
• 55P91: Equivariant homotopy theory, See also 19L47
• 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

Software and tables

Other web sites with this focus

Selected topics at this site

• Under what conditions is an open subset of R^n contractible?
• What should we take for an infinite-dimensional sphere, and is it contractible?
• Complements of compact sets in Hilbert space are contractible.
• What is Homotopy theory all about? [John Baez]
• Invariance of Domain theorems
• Where to find proofs of the Jordan Curve Theorem?
• Proving two curves crossing a square must intersect, using the Brouwer Fixed Point Theorem
• The Ham Sandwich Theorem (Borsuk-Ulam theorem)
• Borsuk-Ulam theorem: no injections from S^n to R^n
• Generalization of the Borsuk-Ulam theorem: no injections from S(k,n) to R^n where S(k,n)=n-skeleton of k-cube
• Are there maps between a space and its loop space? (Not usually)