Geometry Seminar Luigi Bianchi II-1984 by Mauro Meschiari, John H. Rawnsley, Simon Salamon, Edoardo

By Mauro Meschiari, John H. Rawnsley, Simon Salamon, Edoardo Vesentini

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Then deg (sx0 ) = +1 if x0 ∈ U , 0 if x0 ∈ /U . Proof. We use the differential definition of degree. If x0 ∈ / U , then s−1 x0 (0) = ∅, −1 hence deg (sx0 ) = 0. If x0 ∈ U then sx0 (0) = {x0 } and x0 is a regular value with D(sx0 ) = id hence deg (sx0 ) = sgn (D(sx0 )x0 ) = 1. 2. Fixed point index We will use the degree to define the fixed point index which is the algebraic measure of the number of fixed points (cf. 1)) of a continuous selfmap. Let X be a topological space, U its subset and f: U → X a map.

Now the product formula for the degree follows from the commutativity of the diagram Hn+n (U ×U , U ×U \(f ×f )−1 (0)) (f×f )∗ /H n+n (Rn ×Rn , Rn ×Rn \0) =  Hn+n ((U, U \f −1 (0))×(U , U \f −1 (f×f )∗ (0))) =  Hn ((U, U \f −1 (0))⊗Hn (U , U \f /H  n+n = (Rn ×Rn , Rn ×Rn \0) = −1 (0)))  / H (Rn, Rn \0)⊗H (Rn , Rn \0) n n f∗ ⊗f∗ since the homology element representing the orientation z(f×f )−1 (0) ∈ Hn+n (U × U , U × U \ (f × f )−1 (0)) corresponds to zf −1 (0) ⊗ zf −1 (0) ∈ Hn ((U , U \ f −1 (0)) ⊗ Hn (U , U \ f −1 (0))) by the left vertical arrows.

Now ind (f0 ) = deg (p1 − f0 ) = deg (p1 − f1 ) = ind (f1 ). 10) Lemma (Multiplicativity). If f: U → E, f : U → E are compactly fixed, then so is f × f : U × U → E × E and ind (f × f ) = ind (f) · ind (f ). Proof. 13). Now ind (f ×f ) = deg (id ×id’−f ×f ) = deg (id−f) deg (id’−f ) = ind (f)·ind (f ). The fixed point index possesses also a very important property which will enable us to extend its definition to a much larger class of spaces. 11) Lemma (Commutativity Property). Let U ⊂ E, U ⊂ E be open subsets of Euclidean spaces and let f: U → E and g: U → E be continuous maps.

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