(With Thomas Baird) Smoothing maps into algebraic sets and the topological Atiyah-Segal map.
Let X be a real algebraic set in R^n. We show that all continuous maps from M to X, with M a smooth, closed manifold, are homotopic (in X) to smooth maps.
We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations. This leads to an
explanation for certain phenomena in deformation K-theory, and has applications to spaces of flat connections over aspherical manifolds.
(With Rufus Willett and Guoliang Yu) A finite dimensional approach to the strong
Novikov conjecture. Submitted (May 2012).
The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using the Atiyah-Singer families index theorem, and partly by Carlsson's deformation K-theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the K-theory and cohomology of representation spaces.
(With Romain Tessera and Guoliang Yu) Finite decomposition complexity and the
integral Novikov conjecture for higher algebraic K-theory. To appear in J. Reine Angew. Math. (Crelle's Journal).
Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension.
We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity.
This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the K-theoretic assembly map, for groups with
finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all
(geometrically finite) linear groups.
Periodicity in the stable representation theory of crystallographic groups. To appear in Forum Math. (published on-line 11/4/11).
Deformation K-theory associates to each discrete group G a
spectrum built from spaces of finite dimensional unitary
representations of G. In all known examples, the deformation
K-theory spectrum is 2-periodic above the rational cohomological
dimension of the group, in the sense that T. Lawson's Bott map
is an isomorphism on homotopy. In this article, we show that
crystallographic groups G < Isom (R^k) exhibit such
periodicity. Under certain fibering conditions on R^k/G, we
prove a related vanishing result for the homotopy of the stable
moduli space of flat connections over R^k/G, and we provide
examples relating these homotopy groups to the cohomology of G.
These results are established by showing that for each n
> 0, the one-point compactification of the moduli space of
irreducible n-dimensional representations of G is a CW-complex
of dimension at most k. This is proven using results from real
algebraic geometry and projective representation theory.
tubular neighborhoods in infinite-dimensional Riemannian
geometry, with applications to Yang-Mills theory. Arch. Math.
(Basel) 96 (2011) no. 6, 589--599. arXiv:1006.0063
We show that if G is an arbitrary group acting isometrically on
an a (possibly infinite dimensional) Riemannian manifold, then
every G-invariant submanifold with locally trivial normal bundle
admits a G-invariant total tubular neighborhood. These results
apply, in particular, to the Morse strata of the Yang-Mills
functional over a closed surface. The resulting neighborhoods
play an important role in calculations of gauge-equivariant
cohomology for moduli spaces of flat connections over
The Yang-Mills stratification for surfaces. Proc. Amer.
Math. Soc. 139 (2011), no. 5., 1851-1863.
Atiyah and Bott showed that Morse
theory for the Yang-Mills functional can be used to study the
space of flat, or more generally central, connections on a
bundle over a Riemann surface. These methods have recently been
extended to non-orientable surfaces by Ho and Liu. In this
article, we use Morse theory to determine the exact connectivity
of the natural map from
the homotopy orbits of the space of central Yang-Mills
connections to the classifying space of the gauge group. The key
ingredient in this computation is a combinatorial study of the
Morse indices of Yang-Mills critical sets. (This paper was
originally part of a longer paper, which is still available on
stable moduli space of flat connections over a surface. To
appear in Transactions of the AMS. arXiv:0810.4882
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
Nan-Kuo Ho and Chiu-Chu Melissa Liu) Orientability
in Yang-Mills Theory over Nonorientable Surfaces.
in Analysis and Geometry 17 (2009), no. 5, 903--954.
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
for deformation K-theory of free products. Algebraic
& Geometric Topology 7 (2007) 2239--2270.
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.
theory over surfaces and the Atiyah-Segal theorem. Algebraic
& Geometric Topology 8 (2008) 2209--2251.
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.
Representation Theory of Infinite Discrete Groups. This is
the final version of my Ph.D. thesis, posted 7/27/2007.
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
theory over surfaces and the Atiyah-Segal Theorem"]
invariance of deformation K-theory. Posted 10/19/2006.
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
of the coset poset and the subgroup poset of a group. J. Group
Theory 8 (2005), no. 6, 719--746. arXiv:math/0210001
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.
of the coset poset. Senior Honors Thesis, Cornell University
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.