Daniel A. Ramras  

Research Abstracts
  • The stable moduli space of flat connections over a surface. Posted 8/12/2008.

    Abstract: We compute the homotopy type of the moduli space of flat, unitary connections over a compact, aspherical surface, after stabilizing with respect to the rank of the underlying bundle. Over an orientable surface M, we show that this space has the homotopy type of the infinite symmetric product of M, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawson's work on the Bott map in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.

  • (With Nan-Kuo Ho and Chiu-Chu Melissa Liu) Orientability in Yang-Mills Theory over Nonorientable Surfaces." Posted 10/27/2008.

    Abstract: In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of this nonorientable surface. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when the Euler characteristic of the nonorientable surface is negative, and for n<4 when the Euler characteristic of the nonorientable surface is positive (so it is the real projective plane) or zero (so it is the Klein bottle).

  • The Yang-Mills stratification for surfaces revisited. Posted 12/3/07; last updated 5/27/08.

    Abstract: We revisit Atiyah and Bott's study of Morse theory for the Yang--Mills functional over a Riemann surface. We construct gauge-invariant tubular neighborhoods for the Yang-Mills strata and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map Nss(E)//G(E) \to \Map(M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Liu's non-orientable Yang-Mills theory. Our construction of invariant tubular neighborhoods for locally closed submanifolds of infinite dimensional Riemannian manifolds (with an isometric group action) may be of independent interest.

  • Excision for deformation K-theory of free products. Published in Algebraic & Geometric Topology 7 (2007) 2239-2270. Posted on the arXiv 5/15/07, updated 11/28/07.

    Abstract: Associated to a discrete group G, one has the topological category of finite dimensional (unitary) G-representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$-theory spectrum is Carlsson's deformation K-theory \K(G). The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to G*H (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.

  • Yang-Mills theory over surfaces and the Atiyah-Segal theorem. Submitted version, updated 10/28/2008.

    Abstract: In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma) of a compact Lie group \Gamma to the complex K-theory of the classifying space B\Gamma. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation K-theory spectrum \K (\Gamma) (the homotopy-theoretical analogue of R(\Gamma)). Our main theorem provides an isomorphism in homotopy \K^*(\pi_1 \Sigma) = K^*(\Sigma) for all compact, aspherical surfaces \Sigma and all *>0. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.

  • Stable Representation Theory of Infinite Discrete Groups. This is the final version of my Ph.D. thesis, posted 7/27/2007.

    Abstract: The goal of this thesis is to study representations of infinite discrete groups from a homotopical viewpoint. Our main tool and object of study is Carlsson's deformation K-theory, which provides a homotopy theoretical analogue of the classical representation ring. Deformation K-theory is a contravariant functor from discrete groups to connective $\Omega$-spectra, and we begin by discussing a simple model for the zeroth space of this spectrum. We then investigate two related phenomena regarding deformation K-theory: Atiyah-Segal theorems, which relate the deformation $K$-theory of a group to the complex K-theory of its classifying space, and excision, which relates the deformation K-theory of an amalgamation to the deformation K-theory of its factors. In particular, we use Morse theory for the Yang-Mills functional to prove an Atiyah-Segal theorem for fundamental groups of compact, aspherical surfaces, and we prove that deformation K-theory is excisive on all free products. Combined with work of Tyler Lawson, the former result yields homotopical information about the stable coarse moduli space of surface-group representations. [Warning: there are quite a few typographical errors in the main portion of this document, so for material appearing in the papers above, those papers are a much better starting place. Also, one of the arguments in the appendix has been simplified significantly in the appendix to "Yang-Mills theory over surfaces and the Atiyah-Segal Theorem"]

  • Homotopy invariance of deformation K-theory. Posted 10/19/2006.

    • Abstract: This is a brief note describing the starting point of future work on defomation K-theory of equivariant spaces. We extend Carlsson's deformation K-theory to spaces equipped with an action by a discrete monoid, and show that this theory is homotopy invariant under (strong) equivariant homotopy equivalence and under taking products with free monoids. These properties follow from simple arguments analagous to the homotopy invariance of Weibel's homotopy K-theory.

  • Connectivity of the coset poset and the subgroup poset of a group. J. Group Theory 8 (2005), no. 6, 719--746.

    • Abstract: We study the connectivity of the coset poset and the subgroup poset of a group, focusing in particular on simple connectivity. The coset poset was recently introduced by K. S. Brown in connection with the probabilistic zeta function of a group. We take Brown's study of the homotopy type of the coset poset further, and in particular generalize his results on direct products and classify direct products with simply connected coset posets. The homotopy type of the subgroup poset L(G) has been examined previously by Kratzer, Thevenaz, and Shareshian. We generalize some results of Kratzer and Thevenaz, and determine the fundamental group of L(G) in nearly all cases.

  • Connectivity of the coset poset. Senior Honors Thesis, Cornell University (2002). math.GR/0208176

    • Abstract: Topological concepts may be applied to any poset via the simplicial complex of finite chains. The coset poset C(G) of a finite group G (consisting of all cosets of all proper subgroups of G, ordered by inclusion) was introduced by Kenneth S. Brown in his study of the probabilistic zeta function P(G,s), and Brown's results motivate the study of the homotopy type of C(G). After a detailed introduction to basic simplicial topology and homotopy theory for finite posets (Chapter 2), we examine the connectivity of the coset poset (Chapter 3). Of particular interest is the fundamental group, and we provide various conditions under which C(G) is or is not simply connected. In particular, we classify direct products with simply connected coset posets. In Chapter 4, we treat several specific examples, including A_5 and PSL(2,7). We prove that both of these groups have simply connected coset posets, and in fact determine the precise homotopy type of C(A_5). This chapter also includes a detailed determination of all subgroups of the groups PSL(2,p), p prime (following Burnside). The final chapter discusses conjectures and directions for further research.