By Andrew Ranicki

This booklet is an advent to surgical procedure idea: the traditional category process for high-dimensional manifolds. it's geared toward graduate scholars, who've already had a simple topology direction, and could now prefer to comprehend the topology of high-dimensional manifolds. this article includes entry-level money owed of many 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 elemental mechanics of surgical procedure, this ebook comprises many labored examples, invaluable drawings for representation of the algebra and references for additional examining.

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Conversely, if f : C → D is a chain equivalence with chain homotopy inverse g : D → C and chain homotopies h : gf 1, k : f g 1 then the A-module morphisms Γ = 1 0 (−1)i+1 (f h − kf ) 1 k (−1)i g 0 h : C (f )i = Di ⊕ Ci−1 → C (f )i+1 = Di+1 ⊕ Ci define a chain contraction Γ : 0 1 : C (f ) → C (f ) . (iv) If C is any contractible chain complex then H∗ (C) = 0. Conversely, suppose that C is a finite projective A-module chain complex with H∗ (C) = 0. Assume inductively that there exist A-module morphisms Γ : Ci → Ci+1 for i < k such that dC Γ + ΓdC = 1 : Ci → Ci .

Xi−1 , 0, xi , . . , xn−1 ) . The (singular) homology and cohomology groups of X are defined by Hn (X) = Hn (S(X)) = ker(d : Sn (X) → Sn−1 (X))/im(d : Sn+1 (X) → Sn (X)) , H n (X) = H n (S(X)) = ker(d∗ : S n (X) → S n+1 (X))/im(d∗ : S n−1 (X) → S n (X)) with S n (X) = HomZ (Sn (X), Z). For any abelian group G the G-coefficient singular homology groups H∗ (X; G) are defined using the G-coefficient singular chain complex S(X; G) = G ⊗Z S(X) and the G-coefficient singular cohomology is defined using S n (X; G) = HomZ (Sn (X), G) .