The Whitehead Torsion
At first we define the Whitehead torsion for a chain homotopy equivalence of finite based free R-chain complexes. We can assign to the homotopy equivalence its mapping cone C* := cone*(h*) which is a contractible finite based free R-chain complex. Let be any chain contraction of the mapping cone, i.e. for all n. We obtain an isomorphism with, . We define, where A is the matrix of (c* + γ*)odd with respect to the given bases.
For a homotopy equivalence ƒ: X → Y of connected finite CW-complexes we define the Whitehead torsion τ(ƒ) ∈ Wh(π1(Y)) as follows. Let be the lift of ƒ: X → Y to the universal covering. It induces Z-chain homotopy equivalences . Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in which we map to Wh(π1(Y)). This is the Whitehead torsion τ(ƒ) ∈ Wh(π1(Y)).
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“Art is the imposing of a pattern on experience, and our aesthetic enjoyment is recognition of the pattern.”
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