The K\"{u}nneth isomorphism in bounded cohomology preserving the norms
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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 \( B^{\ast}(G) \widehat{\otimes} B^{\ast}(K) \).
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 \( B^{\ast}(G) \widehat{\otimes} B^{\ast}(K) \).
참고문헌 (10)
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