On the Cohomology Rings of Tree Braid Groups
نویسنده
چکیده
Let Γ be a graph. The (unlabelled) configuration space UCnΓ of n points on Γ is the space of n-element subsets of Γ. The n-strand braid group of Γ, denoted BnΓ, is the fundamental group of UCnΓ. We use the methods and results of [10] to get a partial description of the cohomology rings H∗(BnT ), where T is a tree. Our results are then used to prove that BnT is a right-angled Artin group if and only if T is linear or n < 4. This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.
منابع مشابه
Presentations for the Cohomology Rings of Tree Braid Groups
If Γ is a finite graph and n is a natural number, then UCnΓ, the unlabelled configuration space of n points on Γ, is the space of all n-element subsets of Γ. The fundamental group of UCnΓ is the n-strand braid group of Γ, denoted BnΓ. We build on earlier work to compute presentations of the integral cohomology rings H∗(UCnT ;Z), where T is any tree and n is arbitrary. The results suggest that H...
متن کاملOn Rigidity and the Isomorphism Problem for Tree Braid Groups
We solve the isomorphism problem for braid groups on trees with n = 4 or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group BnT on...
متن کاملTopic Proposal The Cohomology of Configuration Space and Representation Stability
The theory of the configuration space of a manifold provides illustrative examples of a number of topological phenomena. We begin with a discussion of the configuration space of the complex plane, and give a classical description of the braid and pure braid groups. We next give an alternative interpretation of the braid groups as mapping classes of punctured disks, an interpretation which provi...
متن کاملRing structures of mod p equivariant cohomology rings and ring homomorphisms between them
In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...
متن کاملCohomology of Pure Braid Groups of exceptional cases
Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...
متن کامل