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Whitehead Torsion


Let (K,L) be a pair consisting of finite, connected CW-complexes where L is a subcomplex of K. Define the associated chain complex C(K,L) group-wise for each p by setting

 C_p(K,L)=H_p(|K^p union L|,|K^(p-1) union L|)
(1)

where H denotes singular homology with integer coefficients and where |K^p| denotes the union of all cells of K of dimension less than or equal to p. Note that C_p is free Abelian with one generator for each p-cell of K-L.

Next, consider the universal covering complexes L^^ subset K^^ of L and K, respectively. The fundamental group pi_1(K) of K can be identified with the group of deck transformations of K^^ so that each sigma in pi_1(K) determines a map

 sigma:(K^^,L^^)->(K^^,L^^)
(2)

which then induces a chain map

 sigma_#:C(K^^,L^^)->C(K^^,L^^).
(3)

The chain map sigma_# turns each chain group C_p(K^^,L^^) into a module over the group ring Zpi_1(K) which is Zpi_1(K)-free with one generator for each p-cell of K-L and which is finitely generated over Zpi_1(K) due to the finiteness of K.

Hence, there is a free chain complex

 C_n(K^^,L^^)->C_(n-1)(K^^,L^^)->...->C_0(K^^,L^^)
(4)

over Zpi_1(K), the homology groups H_i(K^^,L^^) of which are zero due to the fact that |K^^| deformation retracts onto |L^^|. A simple argument shows the existence of a so-called preferred basis for each C_p(K^^,L^^) (Milnor), whereby one can define the Whitehead torsion to be the image of the torsion tauC(K^^,L^^) of the complex C(K^^,L^^) in the Whitehead quotient group Wh(pi_1(K)).

Worth noting is that the Whitehead torsion is an obvious generalization of the Reidemeister torsion, the prior of which is defined to be an Abelian group element rather than an algebraic number like the latter. Experts note that the study of Reidemeister torsion has since been subsumed in the study of Whitehead torsion (Ranicki 1997) whereas Whitehead torsion provides a fundamental tool for the examination of differentiable and combinatorial manifolds having nontrivial fundamental group.


See also

Algebraic Number, Analytic Torsion, Cell, Chain, Chain Complex, Chain Homomorphism, Connected, CW-Complex, Deck Transformation, Deformation Retract, Free Abelian Group, Fundamental Group, Group, Group Generators, Group Ring, Group Torsion, Homology Group, Image, Manifold, Module, Quotient Group, R-Module, Reduced Whitehead Group, Reidemeister Torsion, Singular Homology, Smooth Manifold, Torsion, Union, Universal Cover, Whitehead Group

This entry contributed by Christopher Stover

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References

Milnor, J. "Whitehead Torsion." Bull. Amer. Math. Soc. 72, 358-423, 1966.Ranicki, A. "Notes on Reidemeister Torsion." 1997. http://www.maths.ed.ac.uk/~aar/papers/torsion.pdf.

Cite this as:

Stover, Christopher. "Whitehead Torsion." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WhiteheadTorsion.html

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