Non vanishing of the fourth bounded cohomology of free groups and codimension 2 subspaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908427959140352 |
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| author | Kastenholz, Thorben |
| author_facet | Kastenholz, Thorben |
| contents | In this note we prove that the fouth bounded cohomology of non-abelian free
groups with trivial real coefficients is non-zero. In order to prove this, we
establish a splitting argument whose simplest form is as follows: Let
$M$ denote an $n$-manifold of non-zero simplicial volume and
$S$ a codimension two submanifold of $M$, then one can
conclude that the $n$-th bounded cohomology of the fundamental group of
$M \setminus S$ is non-zero.
While in this note this approach is only used for degree $4$. There is no
reason to expect that this approach and its generalizations is not suitable
to prove the non-vanishing of higher degrees or the bounded cohomology of
different groups as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non vanishing of the fourth bounded cohomology of free groups and codimension 2 subspaces Kastenholz, Thorben Group Theory Algebraic Topology Geometric Topology 20J06, 57M60, 57K40, 57N40 In this note we prove that the fouth bounded cohomology of non-abelian free groups with trivial real coefficients is non-zero. In order to prove this, we establish a splitting argument whose simplest form is as follows: Let $M$ denote an $n$-manifold of non-zero simplicial volume and $S$ a codimension two submanifold of $M$, then one can conclude that the $n$-th bounded cohomology of the fundamental group of $M \setminus S$ is non-zero. While in this note this approach is only used for degree $4$. There is no reason to expect that this approach and its generalizations is not suitable to prove the non-vanishing of higher degrees or the bounded cohomology of different groups as well. |
| title | Non vanishing of the fourth bounded cohomology of free groups and codimension 2 subspaces |
| topic | Group Theory Algebraic Topology Geometric Topology 20J06, 57M60, 57K40, 57N40 |
| url | https://arxiv.org/abs/2503.22511 |