Browsing by Author "Ramras, Daniel A."

Now showing 1 - 6 of 6
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    Commutative cocycles and stable bundles over surfaces
    (De Gruyter, 2019-11) Ramras, Daniel A.; Villareal, Bernardo; Mathematical Sciences, School of Science
    Commutative 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.
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    Extending Properties to Relatively Hyperbolic Groups
    (Duke University, 2019-06) Ramras, Daniel A.; Ramsey, Bobby W.; Mathematical Sciences, School of Science
    Consider 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.
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    Hilbert–Poincaré series for spaces of commuting elements in Lie groups
    (Springer, 2018) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of Science
    In 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.
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    Homological Stability for Spaces of Commuting Elements in Lie Groups
    (Oxford, 2021-03) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of Science
    In 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.
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    The homotopy groups of a homotopy group completion
    (Springer, 2019) Ramras, Daniel A.; Mathematical Sciences, School of Science
    Let 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.
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    A note on orbit categories, classifying spaces, and generalized homotopy fixed points
    (Springer, 2018-03) Ramras, Daniel A.; Mathematical Sciences, School of Science
    We 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.