By Andrew Ranicki

This booklet is an advent to surgical procedure thought: the normal class procedure for high-dimensional manifolds. it really is geared toward graduate scholars, who've already had a easy topology path, and may now wish 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 easy homotopy and homology, Poincare duality, bundles, co-bordism, embeddings, immersions, Whitehead torsion, Poincare complexes, round fibrations and quadratic varieties and formations. whereas targeting the fundamental mechanics of surgical procedure, this e-book contains many labored examples, necessary drawings for representation of the algebra and references for additional studying.

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H n (X) 2 / H n (X) → H n (X; Z2 ) → H n+1 (X) → . . 15. The R-coefficient homology and cohomology groups H∗ (X; R), H ∗ (X; R) are R-modules which are related by evaluation morphisms H n (X; R) → HomR (Hn (X; R), R) ; f → (x → f (x)) . Given an R-module A let T A ⊆ A be the torsion submodule T A = {x ∈ A | sx = 0 ∈ A for some s = 0 ∈ R} . 17 (i) (F -coefficient) For any field F and any n 0 the evaluation morphism e : H n (X; F ) → HomF (Hn (X; F ), F ) ; f → (x → f (x)) is an isomorphism.

1 Let R be a commutative ring. (i) An m-dimensional manifold M is R-orientable if there exists an R-coefficient fundamental class, a homology class [M ] ∈ Hm (M ; R) such that for every x ∈ M the R-module morphism Hm (M ; R) → Hm (M, M \{x}; R) = Hm (Rm , Rm \{0}; R) = R sends [M ] ∈ Hm (M ; R) to a unit in R. (ii) A manifold M is orientable if it is Z-orientable, and nonorientable if it is not Z-orientable. An orientation for an orientable manifold M is a choice of Z-coefficient fundamental class [M ].

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 .

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