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Date: 7-7-2021
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Let be a pair consisting of finite, connected CW-complexes where is a subcomplex of . Define the associated chain complex group-wise for each by setting
(1) |
where denotes singular homology with integer coefficients and where denotes the union of all cells of of dimension less than or equal to . Note that is free Abelian with one generator for each -cell of .
Next, consider the universal covering complexes of and , respectively. The fundamental group of can be identified with the group of deck transformations of so that each determines a map
(2) |
which then induces a chain map
(3) |
The chain map turns each chain group into a module over the group ring which is -free with one generator for each -cell of and which is finitely generated over due to the finiteness of .
Hence, there is a free chain complex
(4) |
over , the homology groups of which are zero due to the fact that deformation retracts onto . A simple argument shows the existence of a so-called preferred basis for each (Milnor), whereby one can define the Whitehead torsion to be the image of the torsion of the complex in the Whitehead quotient group .
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.
REFERENCES:
Milnor, J. "Whitehead Torsion." Bull. Amer. Math. Soc. 72, 358-423, 1966.
Ranicki, A. "Notes on Reidemeister Torsion." 1997. https://www.maths.ed.ac.uk/~aar/papers/torsion.pdf.
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