By Jerzy Jezierski

ISBN-10: 1402039301

ISBN-13: 9781402039300

The concept of a ?xed element performs a very important position in several branches of mat- maticsand its functions. Informationabout the life of such pointsis usually the an important argument in fixing an issue. specifically, topological tools of ?xed aspect idea were an expanding concentration of curiosity over the past century. those topological tools of ?xed aspect thought are divided, approximately conversing, into kinds. The ?rst sort comprises equivalent to the Banach Contraction precept the place the assumptions at the house should be very gentle yet a small switch of the map can get rid of the ?xed aspect. the second one kind, however, corresponding to the Brouwer and Lefschetz fastened aspect Theorems, supply the lifestyles of a ?xed aspect not just for a given map but additionally for any its deformations. This e-book is an exposition of part of the topological ?xed and periodic aspect conception, of this moment kind, in accordance with the notions of Lefschetz and Nielsen numbers. considering either notions are homotopyinvariants, the deformationis used as an important approach, and the assertions of theorems often country the lifestyles of ?xed or periodic issues for each map of the complete homotopy category, we confer with them as homotopy equipment of the topological ?xed and periodic aspect theory.

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**Example text**

We assume that f is an isomorphism. Let us recall that the group Gl(R, n) has two connected components. If the matrix A representing the map f (in a ﬁxed basis) satisﬁes det A > 0, then there is a path in Gl(R, n) joining A with the identity matrix. This yields a homotopy from f to the identity map idE . Now deg (f) = deg (idE ) = 1. If det A < 0, then f can be connected with the map g(t1 , . . , tn ) = (t1 , . . , tn−1 , −tn ). Now by Multiplicity deg (f) = −1. In each case deg (f) = sgn (det (A)).

We deﬁne ind (f) = ind (f| ). 24) Lemma. 1) ind (f) = 0 implies the existence of a ﬁxed point of the map f: D → E. e. fs (x) = x for all x ∈ bd D, then ind (f0 ) = ind (f1 ). Proof. 1) ind (f| ) = ind (f) = 0 implies the existence of a ﬁxed point x ∈ Fix (f| ). This is also a ﬁxed point of f. 2) Since the homotopy {fs } has no ﬁxed point on the boundary, its restriction fs| : int D → E is compactly ﬁxed hence ind (f0| ) = ind (f1| ). It remains to recall that by the deﬁnition ind (fi ) = ind (fi| ) for i = 0, 1.

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. Then the composites gf: V = f −1 (U ) → E, fg: V = g−1 (U ) → E have homeomorphic ﬁxed point sets Fix (gf) = Fix (fg). If moreover these sets are compact, then ind (fg) = ind (gf). Proof. The homeomorphisms are given by the restrictions: f| : Fix (gf) −→ ←− Fix (fg) : g|. In fact if gf(x) = x, then f(x) = f(gf(x)) = fg(f(x)), hence f(x) ∈ Fix(fg). Similarly g(Fix(fg)) ⊂ Fix(gf).

### Homotopy Methods in Topological Fixed and Periodic Points Theory by Jerzy Jezierski

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