Screen LibraryGromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume /
"The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and 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 ac...
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| Main Authors: | |
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| Group Author: | |
| Published: |
American Mathematical Society,
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| Publisher Address: | Providence, RI : |
| Publication Dates: | 2023. |
| Literature type: | Book |
| Language: | English |
| Series: |
Memoirs of the American Mathematical Society,
volume 283, number 1402 (fifth of 7 numbers) |
| Subjects: | |
| Summary: |
"The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and 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 multicomplexes. After setting the main definitions, we construct the singular multicomplex K(X) associated to a topological space X, and we prove that the geometric realization of K(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K(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"-- |
| Carrier Form: | vi, 153 pages : illustrations ; 26 cm. |
| Bibliography: | Includes bibliographical references (pages 149-153). |
| ISBN: |
9781470459918 1470459914 |
| Index Number: | QA169 |
| CLC: | O189.22 |
| Call Number: | O189.22/F912-1 |
| Contents: | Multicomplexes -- The singular multicomplex -- The homotopy theory of complete multicomplexes -- Bounded cohomology of multicomplexes -- The mapping theorem -- The vanishing theorem -- Finiteness and vanishing theorems -- Diffusion of chains -- Admissible submulticomplexes of k(x) -- Diffusion of locally finite chains -- Some results on the simplicial volume of open manifolds. |