By Andrew Ranicki

ISBN-10: 0198509243

ISBN-13: 9780198509240

This ebook is an advent to surgical procedure conception: the traditional category approach for high-dimensional manifolds. it truly is geared toward graduate scholars, who've already had a uncomplicated topology direction, and may now wish to comprehend the topology of high-dimensional manifolds. this article includes entry-level money owed of a number of the must haves of either algebra and topology, together with easy homotopy and homology, Poincare duality, bundles, co-bordism, embeddings, immersions, Whitehead torsion, Poincare complexes, round fibrations and quadratic types and formations. whereas targeting the elemental mechanics of surgical procedure, this publication comprises many labored examples, worthwhile drawings for representation of the algebra and references for extra analyzing.

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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 . (i) If f has no critical points then (W ; M, M ) is a trivial h-cobordism, with a diffeomorphism (W ; M, M ) ∼ = M × (I; {0}, {1}) which is the identity on M .

A CW complex is a space obtained from ∅ by successively attaching cells of non-decreasing dimension X = ( D0 ) ∪ ( D1 ) ∪ ( D2 ) ∪ . . The images of the maps Dn → X are called the n-cells of X. 6 The following conditions on a map f : X → Y of connected CW complexes are equivalent : (i) f is a homotopy equivalence, (ii) f induces isomorphisms f∗ : π∗ (X) → π∗ (Y ), (iii) π∗ (f ) = 0. 2 of Bredon [10]. 7 Let n 1. (i) A space X is n-connected if it is connected and πi (X) = 0 (i n) . (ii) A map f : X → Y of connected spaces is n-connected if f∗ : πi (X) → πi (Y ) is an isomorphism for i < n and f∗ : πn (X) → πn (Y ) is onto, or equivalently if πi (f ) = 0 (i n) .

8 (i) S n is (n − 1)-connected. (ii) (Dn , S n−1 ) is (n − 1)-connected. 9 (i) The suspension of a pointed space X is the pointed space ΣX = S 1 × X/(S 1 × {∗} ∪ {1} × X) , with ∗ ∈ X, 1 ∈ S 1 base points. (ii) The suspension map in the homotopy groups is defined by E : πm (X) → πm+1 (ΣX) ; (f : S m → X) → (Σf : Σ(S m ) = S m+1 → ΣX) . 10 If X is an (n − 1)-connected space for some n 2 then the suspension map E : πm (X) → πm+1 (ΣX) is an isomorphism m < 2n − 1 and a surjection for m = 2n − 1. 13].