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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).