Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(4): 873-890

Online first article February 13, 2020      Printed July 31, 2020

https://doi.org/10.4134/BKMS.b190547

Copyright © The Korean Mathematical Society.

The K\"{u}nneth isomorphism in bounded cohomology preserving the norms

HeeSook Park

Sunchon National University

Abstract

In this paper, for discrete groups \( G \) and \( K \), we show that the cohomology of the complex of projective tensor product \( B^{\ast}(G) \widehat{\otimes} B^{\ast}(K) \) is isomorphic to the bounded cohomology \( \widehat{H}^{\ast}(G \times K) \) of \( G \times K \), which is the cohomology of \( B^{\ast}(G \times K) \) as topological vector spaces, where \(B^{\ast}(G) \) is a complex of bounded cochains of \( G \) with real coefficients \( \mathbb{R} \). In fact, we construct an isomorphism between these two cohomology groups that carries the canonical seminorm in \( \widehat{H}^{\ast}(G \times K) \) to the seminorm in the cohomology of \( B^{\ast}(G) \widehat{\otimes} B^{\ast}(K) \).

Keywords: Bounded cohomology, resolution

MSC numbers: Primary 55T99; Secondary 18G40

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