About: Whitehead torsion

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In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group . These concepts are named after the mathematician J. H. C. Whitehead.

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rdfs:label	Whitehead torsion (en)
rdfs:comment	In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group . These concepts are named after the mathematician J. H. C. Whitehead. (en)
dcterms:subject	Algebraic K-theory Surgery theory Geometric topology
Wikipage page ID	681186 (xsd:integer)
Wikipage revision ID	996541127 (xsd:integer)
Link from a Wikipage to another Wikipage	American Journal of Mathematics Princeton University Press Robion Kirby Elementary matrix Braid group Derived subgroup Algebraic K-theory Algebraic K-theory Homotopy equivalence Hyman Bass Elementary matrices Trivial group Surgery theory Geometric topology Real representation Golden ratio Graham Higman Stephen Smale Fundamental group Main diagonal Publications Mathématiques de l'IHÉS Torus H-cobordism Laurent C. Siebenmann Wall's finiteness obstruction Cyclic group Euclidean domain F. Thomas Farrell Direct limit Handle decomposition Handlebody Mapping cone (homological algebra) Geometric topology J. H. C. Whitehead Surgery theory Cokernel Manifold Manifolds Bulletin of the AMS Piecewise linear manifold Free abelian group Group ring Topological manifold Shear matrix Rational representation University of Tokyo Press Richard Swan Simply connected S-Cobordism theorem S-cobordism theorem Reidemeister torsion Whitney trick Simple homotopy equivalence CW-complex dbr:Proceedings_of_the_London_Mathematical__Society
Link from a Wikipage to an external page	https://arxiv.org/abs/math.GT/0108115 http://www.numdam.org/item%3Fid=PMIHES_1964__22__61_0
sameAs	Whitehead torsion Whitehead torsion Whitehead torsion
dbp:wikiPageUsesTemplate	dbt:Citation
has abstract	In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group . These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non-simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and Laurent C. Siebenmann. The restriction to manifolds of dimension greater than four are due to the application of the Whitney trick for removing double points. In generalizing the h-cobordism theorem, which is a statement about simply connected manifolds, to non-simply connected manifolds, one must distinguish simple homotopy equivalences and non-simple homotopy equivalences. While an h-cobordism W between simply-connected closed connected manifolds M and N of dimension n > 4 is isomorphic to a cylinder (the corresponding homotopy equivalence can be taken to be a diffeomorphism, PL-isomorphism, or homeomorphism, respectively), the s-cobordism theorem states that if the manifolds are not simply-connected, an h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion vanishes. (en)
prov:wasDerivedFrom	wikipedia-en:Whitehead_torsion?oldid=996541127&ns=0
page length (characters) of wiki page	11647 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Whitehead_torsion
is Link from a Wikipage to another Wikipage of	List of algebraic topology topics Nielsen transformation Analysis Situs (book) Analytic torsion Glossary of algebraic topology Stephen Smale H-cobordism K-theory Wall's finiteness obstruction Direct limit of groups Farrell–Jones conjecture J. H. C. Whitehead Surgery theory Homotopy theory Thompson groups Torsion Whitehead group Assembly map Surgery exact sequence Simple-homotopy equivalence Whitehead problem Set-theoretic topology Simple homotopy theory Surgery structure set
is Wikipage disambiguates of	Torsion
is known for of	Stephen Smale
is known for of	Stephen Smale
is foaf:primaryTopic of	wikipedia-en:Whitehead_torsion

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