By L.F. McAuley

ISBN-10: 3540070192

ISBN-13: 9783540070191

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Sp]), thus preserves Zi and Bi . By the deﬁnition (cf. 19) Hi(f): Hi (X; Z) = Zi (X)/Bi (X) → Zi (Y )/Bi (Y ) = Hi(Y ; Z) is the quotient map. e. a Z-linear map, deﬁnes a K -linear map of the linear space over K with the same basis. e. we can multiply elements of K by integers. Using once more the isomorphism K n ≡ Zn ⊗Z K , we can write this homomorphism as φ ⊗Z I|K . By deﬁnition, in the given basis φ and φ ⊗Z I|K have the same matrix with integral coeﬃcients. 20) di ⊗Z I|K Ci (f)⊗Z I|K di ⊗Z I|K Ci−1 (X; Z) ⊗Z K / Ci (X; Z) ⊗Z K Ci−1 (f)⊗Z I|K / Ci−1 (X; Z) ⊗Z K , which follows from the corresponding diagram for Ci(f).

This simplex is the convex set spanned by the vertices of the standard basis of Rn+1 : v0 = (1, 0, . . , 0), v1 = (0, 1, 0, . . , 0), . . , vn = (0, . . , 0, 1). 20 CHAPTER II. LEFSCHETZ–HOPF FIXED POINT THEORY Now any choice of aﬃnely independent points a0 , . . , an determines the aﬃne map φ: ∆n → Rn given by φ(vi ) = ai . If y ∈ int (φ(∆n )), then the homology class [φ] is a generator of the homology group Hn (E, E \ {y}) = Z. Let us denote this generator by zy . Let us ﬁx a local orientation zy0 (a generator of Hn (E, E \ y0 )) near y0 .

Moreover, (id − f)−1 (0) = Fix (f) = Fix (fU ) = (id − f)−1 U (0). 9) the degrees are equal. 7) Lemma Units. Let ρ: U → E be the constant map ρ(U ) = x0 . Then ind (ρ) = +1 if x0 ∈ U , 0 if x0 ∈ /U . Proof. 32)). 8) 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 Fix (f), then ind (f) = ind (f|U1 ) + ind(f|U2 ), Proof. 11), hence ind (f) = deg (id−f) = deg (id−f|U1 )+deg (id−f|U2 ) = ind (f|U1 )+ind (f|U2 ).

### Algebraic and Geometrical Methods in Topology by L.F. McAuley

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