Browsing by Author "Ramras, Daniel A."
Now showing 1 - 8 of 8
Results Per Page
Sort Options
Item Commutative cocycles and stable bundles over surfaces(De Gruyter, 2019-11) Ramras, Daniel A.; Villareal, Bernardo; Mathematical Sciences, School of ScienceCommutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles and are analyzed via the point-wise inversion operation on commutative cocycles.Item Extending Properties to Relatively Hyperbolic Groups(Duke University, 2019-06) Ramras, Daniel A.; Ramsey, Bobby W.; Mathematical Sciences, School of ScienceConsider a finitely generated group G that is relatively hyperbolic with respect to a family of subgroups H1,…,Hn. We present an axiomatic approach to the problem of extending metric properties from the subgroups Hi to the full group G. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.Item Hilbert–Poincaré series for spaces of commuting elements in Lie groups(Springer, 2018) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of ScienceIn this article we study the homology of spaces Hom(Zn,G) of ordered pairwise commuting n-tuples in a Lie group G. We give an explicit formula for the Poincaré series of these spaces in terms of invariants of the Weyl group of G. By work of Bergeron and Silberman, our results also apply to Hom(Fn/Γmn,G) , where the subgroups Γmn are the terms in the descending central series of the free group Fn . Finally, we show that there is a stable equivalence between the space Comm(G) studied by Cohen–Stafa and its nilpotent analogues.Item Homological Stability for Spaces of Commuting Elements in Lie Groups(Oxford, 2021-03) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of ScienceIn this paper, we study homological stability for spaces Hom(Zn,G) of pairwise commuting n-tuples in a Lie group G. We prove that for each n⩾1, these spaces satisfy rational homological stability as G ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, Comm(G) and BcomG, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases. Our proofs use the theory of representation stability—in particular, the theory of FIW-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.Item The homotopy groups of a homotopy group completion(Springer, 2019) Ramras, Daniel A.; Mathematical Sciences, School of ScienceLet M be a topological monoid with homotopy group completion ΩBM. Under a strong homotopy commutativity hypothesis on M, we show that πk(ΩBM) is the quotient of the monoid of free homotopy classes [Sk, M] by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of pointwise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.Item A note on orbit categories, classifying spaces, and generalized homotopy fixed points(Springer, 2018-03) Ramras, Daniel A.; Mathematical Sciences, School of ScienceWe give a new description of Rosenthal’s generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.Item The topological Atiyah-Segal map(arXiv, 2023) Ramras, Daniel A.; Mathematical Sciences, School of ScienceAssociated to each finite dimensional linear representation of a group G, there is a vector bundle over the classifying space BG. This construction was studied extensively for compact groups by Atiyah and Segal. We introduce a homotopy theoretical framework for studying the Atiyah-Segal construction in the context of infinite discrete groups, taking into account the topology of representation spaces. We explain how this framework relates to the Novikov conjecture, and we consider applications to spaces of flat connections on the over the 3-dimensional Heisenberg manifold and families of flat bundles over classifying spaces of groups satisfying Kazhdan's property (T).Item Variations on the nerve theorem(arXiv, 2023) Ramras, Daniel A.; Mathematical Sciences, School of ScienceGiven a locally finite cover of a simplicial complex by subcomplexes, Björner's version of the Nerve Theorem provides conditions under which the homotopy groups of the nerve agree with those of the original complex through a range of dimensions. We extend this result to covers of CW complexes by subcomplexes and to open covers of arbitrary topological spaces, without local finiteness restrictions. Moreover, we show that under somewhat weaker hypotheses, the same conclusion holds when one utilizes the completed nerve introduced by Fernández-Minian. Additionally, we prove homological versions of these results, extending work of Mirzaii and van der Kallen in the simplicial setting. Our main tool is the Čech complex associated to a cover, as analyzed in work of Dugger and Isaksen. As applications, we prove a generalized crosscut theorem for posets and some variations on Quillen's Poset Fiber Theorem.