By Andrew Ranicki

This ebook is an advent to surgical procedure conception: the traditional category approach for high-dimensional manifolds. it truly is aimed toward graduate scholars, who've already had a easy topology direction, and might now wish to comprehend the topology of high-dimensional manifolds. this article comprises entry-level debts of a number of the necessities of either algebra and topology, together with simple homotopy and homology, Poincare duality, bundles, co-bordism, embeddings, immersions, Whitehead torsion, Poincare complexes, round fibrations and quadratic varieties and formations. whereas targeting the elemental mechanics of surgical procedure, this ebook comprises many labored examples, helpful drawings for representation of the algebra and references for extra examining.

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**Extra resources for Algebraic and Geometric Surgery**

**Example text**

6 The Z2 -homology and cohomology of the m-dimensional real projective space RPm = S m /{x ∼ −x} are given by Hi (RPm ; Z2 ) = H m−i (RPm ; Z2 ) = Z2 (0 i m) , and the orientation character is w(RPm ) = (−1)m+1 ∈ H 1 (RPm ; Z2 ) = Z2 . Thus RPm is orientable for odd m, and nonorientable for even m. ✷ There are also versions of Poincar´e duality for manifolds with boundary, and cobordisms. 7 Let R be a commutative ring. (i) An (m + 1)-dimensional cobordism (W ; M, M ) is R-orientable if there exists an R-coefficient fundamental class [W ] ∈ Hm+1 (W, M ∪ M ; R), such that for every x ∈ W \(M ∪ M ) the R-module morphism Hm+1 (W, M ∪M ; R) → Hm+1 (W, W \{x}; R) = Hm+1 (Rm+1 , Rm+1 \{0}; R) = R sends [W ] to a unit of R, and such that ∂([W ]) = ([M ], −[M ]) ∈ Hm (M ∪ M ; R) = Hm (M ; R) ⊕ Hm (M ; R) with [M ] ∈ Hm (M ; R), [M ] ∈ Hm (M ; R) R-coefficient fundamental classes.

Wk Mk (ii) Closed m-dimensional manifolds M, M are cobordant if and only if M can be obtained from M by a sequence of surgeries. 14) any cobordism admits a Morse function f : (W ; M, M ) → I with M = f −1 (0), M = f −1 (1), and such that all the critical values are in the interior of I. Since W is compact there is only a finite number of critical points : label them pj ∈ W (1 j k). Write the critical values as cj = f (pj ) ∈ R, and let ij be the index of pj .

1 establishes Poincar´e duality. 2 of the effect of a surgery on the homotopy and homology groups of a manifold. 3 recalls the classification of surfaces, and gives a complete description of the effects of surgery on surfaces. 4 describes the algebraic properties of rings with involution and sesquilinear forms needed for the intersection form of a non-simply-connected manifold. 5 gives Poincar´e duality for the universal cover of a manifold. 1 Poincar´ e duality The Poincar´e duality isomorphisms H∗ (M ) ∼ = H m−∗ (M ) of an orientable mdimensional manifold M are the global expression of the local property that every x ∈ M has a neighbourhood U ⊆ M which is diffeomorphic to Rm , with H∗ (M, M \{x}) ∼ = H∗ (Rm , Rm \{0}) ∼ = H m−∗ ({0}) ∼ = H m−∗ ({x}) .