Van Kampen’s embedding obstruction for discrete groups

An Obstruction to Embedding Right-angled Artin Groups in Mapping Class Groups

For every orientable surface of finite negative Euler characteristic, we find a rightangled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph Γ to embed into the mapping class group of a surface S, we show that the chromatic number of Γ cannot exceed the chromatic number of the clique graph of the c...

Subnormal Embedding Theorems for Groups

In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into fin...

Van Kampen’s Embedding Obstruction Is Incomplete for 2-complexes in R Michael H. Freedman, Vyacheslav Krushkal and Peter Teichner

In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K ,Z) which vanishes if and only if an n-dimensional simplicial complex K admits a piecewise-linear embedding into R, n ≥ 3. Here K is the deleted product of a complex K. The cohomology in question is the Z/2-equivariant cohomology where Z/2 acts on the space by exchan...

Structural Properties of Inner Amenable Discrete Groups

Higher Minors and Van Kampen’s Obstruction

We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embeddability in the (m−1)-sphere) for H implies its non vanishing for K. As a corollary, based on res...