Whitehead torsion

An element of the reduced Whitehead group $ \overline{K}\; _ {1} A $, constructed from a complex of $ A $- modules. In particular, one has the Whitehead torsion of a mapping complex. Let $ A $ be a ring and let $ F $ be a finitely-generated free $ A $- module. Given two bases $ b = ( b _ {1} \dots b _ {k} ) $ and $ c = ( c _ {1} \dots c _ {k} ) $ of $ F $, if $ c _ {i} = \sum _ {j=} 1 ^ {k} a _ {ij} b _ {j} $, then the matrix $ \| a _ {ij} \| $ is invertible and, hence, defines an element of the group $ \overline{K}\; _ {1} A $, denoted by $ [ c / b ] $. If $ [ c/b ] = 0 $, the bases $ b $ and $ c $ are said to be equivalent. It is clear that

$$ [ e/c ] + [ c/b ] = \ [ e/b ] ,\ [ b/b ] = 0 . $$

For any exact sequence $ 0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0 $ of free $ A $- modules and bases $ e $ of $ E $ and $ g $ of $ G $ one can define a basis $ eg = ( e, f ) $ of $ F $, where the images of the elements $ f $ form the basis $ g $. The equivalence class of this basis depends only on those of $ e $ and $ g $. Now let

$$ C : C _ {n} \mathop \rightarrow \limits ^ \partial C _ {n-} 1 \ \mathop \rightarrow \limits ^ \partial \dots \mathop \rightarrow \limits ^ \partial C _ {0} $$

be a complex of free $ A $- modules $ C _ {i} $ with chosen bases $ c _ {i} $, whose homology complex is free, with a chosen basis $ h _ {i} $. Let the images of the homomorphisms $ \partial : C _ {i+} 1 \rightarrow C _ {i} $ again be free, with basis $ b _ {i} $. The combinations $ b _ {i} h _ {i} b _ {i-} 1 $ define a new basis in $ C _ {i} $. Then the torsion of the complex $ C $ is given by the formula

$$ \tau ( C) = - \sum _ { i= } 0 ^ { n } (- 1) ^ {i} [ c _ {i} / b _ {i} h _ {i} b _ {i-} 1 ] \in \overline{K}\; _ {1} A. $$

This torsion does not depend on the choice of the bases $ b _ {i} $ for the boundary groups but only on $ c _ {i} $ and $ h _ {i} $.

Given a pair $ ( K , L) $ consisting of a finite connected complex $ K $ and a subcomplex $ L $ which is a deformation retract of $ K $, one puts $ \Pi \simeq \pi _ {1} ( K) \simeq \pi _ {1} ( L) $. If $ \widetilde{K} $ and $ \widetilde{L} $ are the universal covering complexes for $ K $ and $ L $, then $ \sigma \in \Pi $ defines a chain mapping $ \sigma : ( \widetilde{k} , \widetilde{i} ) \rightarrow ( \widetilde{K} , \widetilde{L} ) $ and hence a mapping of chain groups $ \sigma _ {*} : C ( \widetilde{K} , \widetilde{L} ) \rightarrow C ( \widetilde{K} , \widetilde{L} ) $, i.e. $ C _ {p} ( \widetilde{K} , \widetilde{L} ) $ is a $ \mathbf Z [ \Pi ] $- module. One thus obtains a free chain complex

$$ C _ {n} ( \widetilde{K} , \widetilde{L} ) \rightarrow C _ {n-} 1 ( \widetilde{K} , \widetilde{L} ) \rightarrow \dots \rightarrow C _ {0} ( \widetilde{K} , \widetilde{L} ) $$

over $ \mathbf Z [ \Pi ] $. The homology of this complex is trivial, i.e. $ \widetilde{L} $ is a deformation retract of $ \widetilde{K} $.

Let $ e _ {1} \dots e _ \alpha $ be $ p $- chains in $ K \setminus L $. For each chain $ e _ {i} $ one chooses a representative $ \widetilde{e} _ {i} $ in $ \widetilde{K} $ lying above $ e _ {i} $ and fixes its orientation. Then $ c _ {p} = ( \widetilde{e} _ {1} \dots \widetilde{e} _ \alpha ) $ is a basis in $ C _ {p} ( \widetilde{K} , \widetilde{L} ) $; hence there is defined a subset $ \tau C ( \widetilde{K} , \widetilde{L} ) $ of $ \widetilde{K} _ {1} \mathbf Z [ \Pi ] $, called the torsion. In general it depends on the choice of the bases $ c _ {p} $. However, the image of this set in the Whitehead group $ \mathop{\rm Wh} ( \Pi ) $ consists of a single element $ \tau ( K, L) $, called the Whitehead torsion of the pair $ ( K , L) $.

An important property of the Whitehead torsion is its combinatorial invariance. Whether $ \tau ( K, L) $ is a topological invariant is not known (1984).

Let $ f: X \rightarrow Y $ be a homotopy equivalence ( $ X, Y $ chain complexes). Then the torsion of the mapping $ f $ is defined as $ \tau ( f ) = f _ {*} \tau ( M _ {f} , X) \in \mathop{\rm Wh} ( \pi _ {1} Y) $, where $ M _ {f} $ is the mapping cylinder of $ f $. If $ \tau ( f ) = 0 $, then $ f $ is called a simple homotopy equivalence. Properties of the torsion $ \tau ( f ) $ are: 1) if $ i : L \rightarrow K $ is an inclusion, then $ \tau ( i) = \tau ( K , L) $; 2) $ \tau ( g \circ f ) = \tau ( g) + g _ {*} \tau ( f ) $; 3) if $ f $ is homotopic to $ f ^ { \prime } $, then $ \tau ( f ) = \tau ( f ^ { \prime } ) $; 4) if $ I $ is the identity mapping of a simply-connected complex with Euler characteristic $ \chi $, then $ \tau ( I \times f ) = \chi \cdot \tau ( f ) $.

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 $ \tau ( K, L) $ 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