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.
simple homotopy equivalence is its Whiteheadtorsion τ ( f ) {\displaystyle \tau (f)} which is an element in the Whitehead group Wh ( π 1 ( Y ) ) {\displaystyle...
torsion, de Rham torsion, Ray-Singer torsion), a topological invariant of manifolds Whiteheadtorsion, in geometric topology Torsion field (pseudoscience)...
can be used to classify lens spaces. Reidemeister torsion is closely related to Whiteheadtorsion; see (Milnor 1966). It has also given some important...
abelian group Whiteheadtorsion List of statements undecidable in ZFC Statements true in L Shelah, S. (1974). "Infinite Abelian groups, Whitehead problem and...
complex or PL-manifold A, equal to Wh(π1(A)); see Whiteheadtorsion. All named after J. H. C. Whitehead. This disambiguation page lists mathematics articles...
Farrell–Jones conjecture. It is even unknown if the Whitehead group of F (see Whiteheadtorsion) or the projective class group of F (see Wall's finiteness...
T(A)} is its torsion subgroup, then the factor group A / T ( A ) {\displaystyle A/T(A)} is torsion-free. However, in general the torsion subgroup is not...
coefficient theorem See also: Characteristic class, Postnikov tower, Whiteheadtorsion There are several specific theories simple homotopy theory stable...
Regular homotopy Sard's theorem Sphere eversion Structural stability Whiteheadtorsion Diffeomorphism Awards Wolf Prize (2007) National Medal of Science...
the corresponding manifold. s-cobordism theorem h-cobordism theorem Whiteheadtorsion Dehn surgery Manifold decomposition Orientation character Plumbing...
with: smooth (DIFF), PL, or topological manifolds and whether we take Whiteheadtorsion into account or not (decorations s {\displaystyle s} or h {\displaystyle...
structure set depending on the category (DIFF, PL or TOP) and whether Whiteheadtorsion is taken into account or not. Let X be a closed smooth (or PL- or...
type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy types". WhiteheadtorsionWhitehead 1950 Cohen, M. M. (1973). A Course...
Moriah, Yoav (1993), "Generating systems of groups and Reidemeister-Whiteheadtorsion", Journal of Algebra, 157 (1): 170–198, doi:10.1006/jabr.1993.1096...
homotopy equivalence is a simple-homotopy equivalence if and only if its Whiteheadtorsion vanishes. simplicial approximation See simplicial approximation theorem...
of seminars and public lectures across Australia. In 2020, she won a Whitehead Prize of the London Mathematical Society "for her deep contributions to...
Jeanne; Mitchell, Laura E; Pasquariello, Patrick S; Sutton, Leslie N; Whitehead, Alexander S (2004). "Spina bifida". The Lancet. 364 (9448): 1885–1895...
double wishbone suspension with a transverse leaf spring in front and a torsion bar in the rear which was upgraded to a de Dion tube for 1950. Worm and...
trivial (since π 2 n − 1 ( S n ) {\displaystyle \pi _{2n-1}(S^{n})} is torsion). If n {\displaystyle n} is even, the image of h {\displaystyle h} contains...
The essential dimension of spin groups is OEIS:A280191. Grothendieck's "torsion index" is OEIS:A096336. Karoubi, Max (2008). K-Theory. Springer. pp. 210–214...
Global Information
