Algebraic and Geometric Surgery by Andrew Ranicki

By Andrew Ranicki

This booklet is an advent to surgical procedure idea: the normal type procedure for high-dimensional manifolds. it truly is aimed toward graduate scholars, who've already had a uncomplicated topology direction, and could now prefer to comprehend the topology of high-dimensional manifolds. this article comprises entry-level bills of some of the must haves 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 types and formations. whereas focusing on the fundamental mechanics of surgical procedure, this ebook contains many labored examples, helpful drawings for representation of the algebra and references for additional studying.

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20 Let f : W m+1 → I be a Morse function on an (m + 1)dimensional manifold cobordism (W ; M, M ) with f −1 (0) = M , f −1 (1) = M , and such that all the critical points of f are in the interior of W .

20 Let f : W m+1 → I be a Morse function on an (m + 1)dimensional manifold cobordism (W ; M, M ) with f −1 (0) = M , f −1 (1) = M , and such that all the critical points of f are in the interior of W .

Ii) If n 2 and X is an (n − 1)-connected space then πn (X) → Hn (X) is an isomorphism of abelian groups. (iii) If n 2 and f : X → Y is an (n − 1)-connected map then πn (f ) → Hn (f ) is an isomorphism of Z[π1 (X)]-modules, with f : X → Y a π1 (X)-equivariant lift of f to the universal covers X, Y of X, Y . 7 of Bredon [10]. 27 (i) A connected space X is n-connected if and only if π1 (X) = {1} and Hi (X) = 0 for 1 i < n. (ii) A map of connected spaces f : X → Y is n-connected if and only if f∗ : π1 (X) → π1 (Y ) is an isomorphism and Hi (f ) = 0 for i n.

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