By Robert F. Brown, Massimo Furi, L. Gorniewicz, Boju Jiang

This ebook is the 1st on the earth literature proposing all new tendencies in topological mounted aspect idea. beforehand all books attached to the topological mounted element concept have been dedicated purely to a couple components of this idea. This booklet might be particularly invaluable for post-graduate scholars and researchers drawn to the fastened aspect thought, rather in topological tools in nonlinear research, differential equations and dynamical structures. The content material is additionally prone to stimulate the curiosity of mathematical economists, inhabitants dynamics specialists in addition to theoretical physicists exploring the topological dynamics.

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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les buildings 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.

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**Example text**

Let us remark that Lefschetz formulated his theorem for compact manifolds. In 1967, A. Granas extended the Lefschetz ﬁxed point theorem to the case of absolute neighbourhood retracts. The proof of the theorem was based on the fact that all compact absolute neighbourhood retracts are homotopically equivalent with polyhedra. Then the case of noncompact absolute neighbourhood retracts was reduced to the compact case by using the generalized trace theory introduced by J. Leray. In the present paper we would like to present current results concerning this theorem for metric spaces.

In order to do this, we will need to know f1∗ : H 1 (S 1 H (S1 ; Γ1 ) for Γ the trivial local coeﬃcient system Q, and the coeﬃcient system given by the local orientation with rational coeﬃcients, denoted by ΓSi . We will present the details for the case where S2 is the Klein bottle and state the results for the other cases. Since f1 is orientation true, it follows that f1 induces a homomorphism in cohomology with the local coeﬃcient system given by local orientation (compare with Section 4). The reader may ﬁnd more details about cohomology with local coeﬃcients in [Wh].

So the coincidence question is equivalent to the ﬁxed point question. 12) Theorem. B(g, g, 1, 1) = 1 − χ(S Sg ) = 4g − 3. The proof of the above result contains two parts. The ﬁrst is to show that B(g, g, 1, 1) ≤ 1 − χ(S Sg ) = 4g − 3. This is done in [Ji8]. The second part is obtained by constructing an example which contains a Nielsen class whose index is 4g − 3. This is Example 2 in [GJi]. 13) Theorem. B(g, g, 1, 0) = 1 − χ(S Sg ) = 2g − 1. The inequality B(g, g, 1, 0) ≤ 1−χ(S Sg ) = 2g−1 follows from [GJi, Proposition 5] and the fact that the inequality is sharp follows from [GJi, Example 4].