Zusammenfassung
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume , bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov's theory of bounded cohomology of topological spaces was based ...
Zusammenfassung
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume , bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov's theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets.The first aim of this paper is to lay the foundation of the theory of multicom-plexes. After setting the main definitions, we construct the singular multicomplex 1C(X) associated to a topological space X, we prove that the geometric real-ization of 1C(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which, roughly speaking, trans-lates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that 1C(X) is complete for every CW complex X.In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov's Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
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