Induccion en la Geometria by Golovina and Yaglom

By Golovina and Yaglom

L. a. INDUCCION EN GEOMETRIA de I. L. GOLOVINA

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We have called [X] a fundamental class, and X a Poincar´e complex, with formal dimension n, when t [X] ∩ : H r (X) → Hn−r (X) is an isomorphism for all r. The map ξ ∩ is then a chain homotopy equivalence. Note that a change in our choices (of the representative cycle ξ and of the chain approximation to the diagonal map) will only affect ξ ∩ by a chain homotopy. We now need the theory of Whitehead torsion. The most convenient account for our purposes is Milnor’s survey article [M14] (other references will be found there).

Note that this condition implies that the i∗ ω π1 (Y ) → {±1}. If Y is not homomorphism w for X is the composite π1 (X) → connected, we require this of each component. (Note that X = ∅ is permitted). The following result shows why this notion will be important for our subsequent investigations. Note that a smooth manifold gives rise (by smooth triangulation) to an essentially unique P L manifold, and that a compact P L manifold has P L triangulations, any two of which admit a common subdivision.

Proof As usual, we may suppose φ an inclusion. Then we have the commutative diagram of based short exact sequences of based chain complexes of finite total rank 0o C ∗ (ψ) o C ∗ (N ) o C ∗ (Y ) o 0 [N ] ∩ − 0  G C∗ (N, M ) [Y ] ∩ −  G C∗ (Y, X) G C∗ (φ) G0 2. simple poincar´ e complexes 29 in which, by hypothesis, the vertical maps are simple equivalences. Now replace C∗ (N, M ) by a suitable equivalent complex such that C∗′ (N, M ) → C∗ (Y, X) is surjective, with kernel D∗ (φ), say this is the chain complex with homology groups K∗ (N, M ) .

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