Read e-book online Combinatorial Mathematics VII. Proc. conf. Newcastle, 1979 PDF

By R. W. Robinson, G. W. Southern, W. D. Wallis

ISBN-10: 354010254X

ISBN-13: 9783540102540

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Extra info for Combinatorial Mathematics VII. Proc. conf. Newcastle, 1979

Example text

Proof. Since X is compact, we can replace W with a compact ∂-manifold W that contains X in its interior. Now W is an ANR, so it is n-LC at x for every n and for every x. The compactness of W together with a Lebesgue number argument establishes the following uniform version of local connectivity: for each nonnegative integer n and for each > 0 there exists δ > 0 such that any map of ∂I n into a δ-subset of W extends to a map of I n into an -subset of W . Let > 0 be given. Choose δk such that any map of ∂I k into a δk -subset of W extends to a map of I k into an ( /2)-subset of W .

4 (Estimated Homotopy Extension Theorem). Let Y be an ANR, X a normal space, f : X → Y a map, b : Y → (0, ∞] another map, A a closed subset of X, U a neighborhood of A in X, and µ : A × I → Y a homotopy such that µ0 = f |A and diam µ({a} × I) < b(µ(a, t)) for all a ∈ A and t ∈ I. Then there exists a homotopy H : X × I → Y such that H0 = f , H|A × I = µ, H({x} × I) = f (x) for all x ∈ X U , and diam H({x} × I) < b(H(x, t)) for all x ∈ X and t ∈ I. Proof. Define a map F on Z = (X × {0}) ∪ (A × I) ⊂ X × I as f on X ×{0} and µ on A×I.

Taking products with Rn−k−1 , one recovers Rn as × S k /R, Rn = (Rn−k−1 × R1+ × S k )/Gk = Rn−k + × S k whose nondegenerate where R now denotes the decomposition of Rn−k + k n−k−1 × {0}. 7. 7. Viewing Rn as the k-spin of Rn−k + Similarly, upon taking one-point compactifications, one can view S n as the quotient space S n = (B n−k × S k )/Tk , where Tk denotes the decomposition of B n−k × S k into points and the k-spheres {x} × S k , x ∈ ∂B n−k . Here 38 1. 8. 8. Viewing S n as the k-spin of B n−k Let π : B n−k × S k → S n = (B n−k × S k )/Tk denote the quotient map.

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Combinatorial Mathematics VII. Proc. conf. Newcastle, 1979 by R. W. Robinson, G. W. Southern, W. D. Wallis


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