Download Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A. PDF

By Hatcher A.

Show description

Read or Download Algebraic topology. Errata (web draft, Nov. 2004) PDF

Similar geometry and topology books

Introduction a la Topologie

Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les constructions permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de los angeles compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

Spaces of Constant Curvature

This ebook is the 6th variation of the vintage areas of continuing Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among team conception and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration concept of finite teams, and of symptoms of contemporary growth in discrete subgroups of Lie teams.

Extra info for Algebraic topology. Errata (web draft, Nov. 2004)

Example text

Homologic definition of the degree. Now we are in a position to present the homologic definition of the degree of a d-compact map. Let E denote n-dimensional Euclidean space, U ⊂ E its open subset. Let us fix an orientation of the space E by fixing a generator z0 ∈ Hn (E, E \ 0). Let f: U → E be a d-compact map. Let zf −1 (0) ∈ Hn (U , U \ f −1 (0)) = Hn (E, E \ f −1 (0)) denote the orientation along the compact set f −1 (0). The map f induces the homomorphism f∗ : Hn (U , U \ f −1 (0)) → Hn (E, E \ 0) = Z.

The diagram Hn (V , V \ f −1 (0)) f∗ = i∗  Hn (U , U \ f −1 (0)) is commutative. / Hn (E, E \ 0) f∗  / Hn (E, E \ 0) 22 CHAPTER II. 19) Lemma (Units). Let i: U → E be the inclusion. Then deg (i) = 1 if 0 ∈ U , 0 if 0 ∈ /U . Proof. We notice that the composition i i ∗ ∗ Hn (E, E \ 0) ←− Hn (U , U \ 0) −→ Hn (E, E \ 0) is the identity map (by excision) if 0 ∈ U and is zero if 0 ∈ /U since then Hn (U , U \ 0) = 0 . 20) Lemma (Additivity). If U 1 , U 2 ⊂ U are open subsets such that the restrictions f|U1 , f|U2 are compactly fixed and U 1 ∩ U 2 is disjoint from f −1 (0), then deg (f) = deg (f|U1 ) + deg (f|U2 ).

21) Lemma (Homotopy Invariance). Let U ⊂ E be an open subset and let F : U × I → E be a d-compact map. Then deg (f0 ) = deg (f1 ), where ft = F ( · , t) for 0 ≤ t ≤ 1. Proof. Since F −1 (0) is compact, p1 (F −1 (0)) ⊂ U is also compact, where p1 : U × I → U is the projection. Now we have a commutative diagram Hn (U , U \ p1 F −1 (0)) =  Hn ((U , U \ p1 F −1 (0)) × I) O = Hn (U , U \ p1 F −1 (0)) / Hn (U , U \ f −1 (0)) 0 f0∗ = i0∗  / Hn (U × I, U × I \ F −1 (0)) O F∗  / Hn (E, E \ 0) O = ii∗ / Hn (U , U \ f −1 (0)) 1 / Hn (E, E \ 0) f1∗ / Hn (E, E \ 0) It remains to notice that i0∗ (zf −1 (0) ) = i1∗ (zf −1 (0) ) = j∗ (zp1 F −1 (0)) ∈ Hn (U × I, 0 1 U × I \ F −1 (0)), where j: (U , U \ p1 F −1 (0)) × I) → (U × I, U × I \ F −1 (0)) is the inclusion and zp1 F −1 (0) ∈ Hm (U , U \ p1 F −1 (0)) = Hm ((U , U \ p1 F −1 (0)) × I).

Download PDF sample

Rated 4.73 of 5 – based on 39 votes