Whitehead torsion

An element of the reduced Whitehead group , constructed from a complex of -modules. In particular, one has the Whitehead torsion of a mapping complex. Let be a ring and let be a finitely-generated free -module. Given two bases and of , if , then the matrix is invertible and, hence, defines an element of the group , denoted by . If , the bases and are said to be equivalent. It is clear that

For any exact sequence of free -modules and bases of and of one can define a basis of , where the images of the elements form the basis . The equivalence class of this basis depends only on those of and . Now let

be a complex of free -modules with chosen bases , whose homology complex is free, with a chosen basis . Let the images of the homomorphisms again be free, with basis . The combinations define a new basis in . Then the torsion of the complex is given by the formula

This torsion does not depend on the choice of the bases for the boundary groups but only on and .

Given a pair consisting of a finite connected complex and a subcomplex which is a deformation retract of , one puts . If and are the universal covering complexes for and , then defines a chain mapping and hence a mapping of chain groups , i.e. is a -module. One thus obtains a free chain complex

over . The homology of this complex is trivial, i.e. is a deformation retract of .

Let be -chains in . For each chain one chooses a representative in lying above and fixes its orientation. Then is a basis in ; hence there is defined a subset of , called the torsion. In general it depends on the choice of the bases . However, the image of this set in the Whitehead group consists of a single element , called the Whitehead torsion of the pair .

An important property of the Whitehead torsion is its combinatorial invariance. Whether is a topological invariant is not known (1984).

Let be a homotopy equivalence ( chain complexes). Then the torsion of the mapping is defined as , where is the mapping cylinder of . If , then is called a simple homotopy equivalence. Properties of the torsion are: 1) if is an inclusion, then ; 2) ; 3) if is homotopic to , then ; 4) if is the identity mapping of a simply-connected complex with Euler characteristic , then .

References

[1]	J.H.C. Whitehead, "Simple homotopy types" Amer. Math. J. , 72 (1950) pp. 1–57
[2]	J.W. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–426

Comments

The topological invariance of is treated in [a1]–[a3].

References

[a1]	T.A. Chapman, "Topological invariance of Whitehead torsion" Amer. J. Math. , 96 (1974) pp. 488–497
[a2]	S. Ferry, "The homeomorphism group of a compact Hilbert cube manifold is an ANR" Ann. of Math. , 106 (1977) pp. 101–119
[a3]	J.E. West, "Mapping Hilbert cube manifolds to ANR's: a solution to a conjecture of Borsuk" Ann. of Math. , 106 (1977) pp. 1–18