Download e-book for iPad: Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983 by Eduard Casas-Alvero, Gerald E. Welters, Sebastian

By Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps

ISBN-10: 3540152326

ISBN-13: 9783540152323

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7. The spaces R, R+ , and I are contractible. 6) Rn , Rn+ and B n are contractible. If p ∈ S n , then S n − {p} is homeomorphic to Rn ; hence S n − {p} is contractible. The subspace A ⊂ X is a retract of X if there is a map r : X → A such i r that A →X −→A is the identity. Such a map r is a retraction of X to A. The subspace A ⊂ X is a strong deformation retract of X if there is a retraction r : X → A and a homotopy F : X × I → X relative to A such that F0 = idX and F1 = i ◦ r. Such a homotopy F is a strong deformation retraction of X to A.

For an arbitrary CW complex X, U ⊂ X is open iff U ∩ X n is open in X n for all n, iff U ∩ ei is open in ei for every i-cell ei of X such that i ≤ n and for every n, iff U ∩ e is open in e for every cell e of X. 13. A compact subset of a CW complex lies in the union of finitely many cells. In particular, a CW complex is finite iff its underlying space is compact. Proof. Suppose this were false. Then there would be a compact subset C of ◦ X such that C ∩ eα = ∅ for infinitely many values of α (where {eα | α ∈ A} is ◦ the set of cells of X).

1 A graded R-module is a sequence C := {Cn }n∈Z of R-modules. If C and D are graded R-modules, a (graded) homomorphism of degree d from C to D is a sequence f := {fn : Cn → Dn+d }n∈Z of R-module homomorphisms. A chain complex over R is a pair (C, ∂) where C is a graded R-module and ∂ : C → C is a homomorphism of degree −1 such that ∂ ◦ ∂ = 0. ∂ is the boundary operator of C. 1 At various points we will explicitly assume more about R, namely that it is either a principal ideal domain (PID) or a field.

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Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983 by Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps


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