698 research outputs found
Morse theory, graphs, and string topology
In these lecture notes we discuss a body of work in which Morse theory is
used to construct various homology and cohomology operations. In the classical
setting of algebraic topology this is done by constructing a moduli space of
graph flows, using homotopy theoretic methods to construct a virtual
fundamental class, and evaluating cohomology classes on this fundamental class.
By using similar constructions based on "fat" or ribbon graphs, we describe how
to construct string topology operations on the loop space of a manifold, using
Morse theoretic techniques. Finally, we discuss how to relate these string
topology operations to the counting of J - holomorphic curves in the cotangent
bundle. We end with speculations about the relationship between the absolute
and relative Gromov-Witten theory of the cotangent bundle, and the open-closed
string topology of the underlying manifold.Comment: 36 pages, 12 figure
The Floer homotopy type of the cotangent bundle
Let M be a closed, oriented, n-dimensional manifold. In this paper we
describe a spectrum in the sense of homotopy theory, Z(T^*M), whose homology is
naturally isomorphic to the Floer homology of the cotangent bundle, T^*M. This
Floer homology is taken with respect to a Hamiltonian H: S^1 x T^*M --> R,
which is quadratic near infinity. Z(T^*M) is constructed assuming a basic
smooth gluing result of J-holomorphic cylinders. This spectrum will have a C.W
decomposition with one cell for every periodic solution of the equation defined
by the Hamiltonian vector field X_H. Its induced cellular chain complex is
exactly the Floer complex. The attaching maps in the C.W structure of Z(T^*M)
are described in terms of the framed cobordism types of the moduli spaces of J
-holomorphic cylinders in T^*M with given boundary conditions. This is done via
a Pontrjagin-Thom construction, and an important ingredient in this is proving,
modulo this gluing result, that these moduli spaces are compact, smooth, framed
manifolds with corners. We then prove that Z(T^*M), which we refer to as the
"Floer homotopy type" of T^*M, has the same homotopy type as the suspension
spectrum of the free loop space, LM. This generalizes the theorem first proved
by C. Viterbo that the Floer homology of T^*M is isomorphic to H_*(LM).Comment: 36 pages. A gluing assumption is described, and a more complete
discussion of framing issues is give
Stability phenomena in the topology of moduli spaces
The recent proof by Madsen and Weiss of Mumford's conjecture on the stable
cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an
important stability theorem about the topology of moduli spaces. In this
article we give a survey of families of classifying spaces and moduli spaces
where "stability phenomena" occur in their topologies. Such stability theorems
have been proved in many situations in the history of topology and geometry,
and the payoff has often been quite remarkable. In this paper we discuss
classical stability theorems such as the Freudenthal suspension theorem, Bott
periodicity, and Whitney's embedding theorems. We then discuss more modern
examples such as those involving configuration spaces of points in manifolds,
holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the
stable topology of general linear groups, and pseudoisotopies of manifolds. We
then discuss the stability theorems regarding the moduli spaces of Riemann
surfaces: Harer's stability theorem on the cohomology of moduli space, and the
Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We
also describe Galatius's recent theorem on the stable cohomology of
automorphisms of free groups. We end by speculating on the existence of general
conditions in which one might expect these stability phenomena to occur.Comment: typos and some references corrected. To appear in "Surveys in
Differential Geometry", vol. on "Geometry of Riemann surfaces and their
moduli spaces
Morse field theory
In this paper we define and study the moduli space of metric-graph-flows in a
manifold M.
This is a space of smooth maps from a finite graph to M, which, when
restricted to each edge, is a gradient flow line of a smooth (and generically
Morse) function on M. Using the model of Gromov-Witten theory, with this moduli
space replacing the space of stable holomorphic curves in a symplectic
manifold, we obtain invariants, which are (co)homology operations in M. The
invariants obtained in this setting are classical cohomology operations such as
cup product, Steenrod squares, and Stiefel-Whitney classes. We show that these
operations satisfy invariance and gluing properties that fit together to give
the structure of a topological quantum field theory. By considering equivariant
operations with respect to the action of the automorphism group of the graph,
the field theory has more structure. It is analogous to a homological conformal
field theory. In particular we show that classical relations such as the Adem
relations and Cartan formulae are consequences of these field theoretic
properties.
These operations are defined and studied using two different methods. First,
we use algebraic topological techniques to define appropriate virtual
fundamental classes of these moduli spaces. This allows us to define the
operations via the corresponding intersection numbers of the moduli space.
Secondly, we use geometric and analytic techniques to study the smoothness and
compactness properties of these moduli spaces. This will allow us to define
these operations on the level of Morse-Smale chain complexes, by appropriately
counting metric-graph-flows with particular boundary conditions.Comment: 59 pages, 10 figure
Twisted Calabi-Yau ring spectra, string topology, and gauge symmetry
In this paper, we import the theory of "Calabi-Yau" algebras and categories
from symplectic topology and topological field theories to the setting of
spectra in stable homotopy theory. Twistings in this theory will be
particularly important. There will be two types of Calabi-Yau structures in the
setting of ring spectra: one that applies to compact algebras and one that
applies to smooth algebras. The main application of twisted compact Calabi-Yau
ring spectra that we will study is to describe, prove, and explain a certain
duality phenomenon in string topology. This is a duality between the manifold
string topology of Chas-Sullivan and the Lie group string topology of
Chataur-Menichi. This will extend and generalize work of Gruher. Then,
generalizing work of the first author and Jones, we show how the gauge group of
the principal bundle acts on this compact Calabi-Yau structure, and compute
some explicit examples. We then extend the notion of the Calabi-Yau structure
to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles
over the loop space, , have this structure. In the case when is a
sphere we will use these twisted smooth Calabi-Yau ring spectra to study
Lagrangian immersions of the sphere into its cotangent bundle. We recast the
work of Abouzaid-Kragh to show that the topological Hochschild homology of the
Thom ring spectrum induced by the -principle classifying map of the
Lagrangian immersion, detects whether that immersion can be Lagrangian isotopic
to an embedding. We then compute some examples. Finally, we interpret these
Calabi-Yau structures directly in terms of topological Hochschild homology and
cohomology
A polarized view of string topology
Let M be a closed, connected manifold, and LM its loop space. In this paper
we describe closed string topology operations in h_*(LM), where h_* is a
generalized homology theory that supports an orientation of M. We will show
that these operations give h_*(LM) the structure of a unital, commutative
Frobenius algebra without a counit. Equivalently they describe a positive
boundary, two dimensional topological quantum field theory associated to
h_*(LM). This implies that there are operations corresponding to any surface
with p incoming and q outgoing boundary components, so long as q >0. The
absence of a counit follows from the nonexistence of an operation associated to
the disk, D^2, viewed as a cobordism from the circle to the empty set. We will
study homological obstructions to constructing such an operation, and show that
in order for such an operation to exist, one must take h_*(LM) to be an
appropriate homological pro-object associated to the loop space. Motivated by
this, we introduce a prospectrum associated to LM when M has an almost complex
structure. Given such a manifold its loop space has a canonical polarization of
its tangent bundle, which is the fundamental feature needed to define this
prospectrum. We refer to this as the "polarized Atiyah - dual" of LM . An
appropriate homology theory applied to this prospectrum would be a candidate
for a theory that supports string topology operations associated to any
surface, including closed surfaces.Comment: final version to appear in, "Topology, Geometry, and Quantum Field
Theory", proceedings of the 2002 Oxford symposium in honour of the 60th
birthday of Graeme Segal, LMS Lecture Note Series 24 pages, 13 figure
Fourier Decompositions of Loop Bundles
In this paper we investigate bundles whose structure group is the loop group
LU(n). Our main result is to give a necessary and sufficient criterion for
there to exist a Fourier type decomposition of such a bundle . This is
essentially a decomposition of as , where
is a finite dimensional subbundle of and is the loop
space of the complex numbers. The criterion is a reduction of the structure
group to the finite rank unitary group U(n) viewed as the subgroup of LU(n)
consisting of constant loops. Next we study the case where is the loop
space of an dimensional bundle . The tangent bundle of is
such a bundle. We then show how to twist such a bundle by elements of the
automorphism group of the pull back of over via the map
that evaluates a loop at a basepoint. Given a connection on , we view
the associated parallel transport operator as an element of this gauge group
and show that twisting the loop bundle by such an operator satisfies the
criterion and admits a Fourier decomposition.Comment: 14 pages, 0 figure
A Morse theoretic description of string topology
Let M be a closed, oriented, n-dimensional manifold. In this paper we give a
Morse theoretic description of the string topology operations introduced by
Chas and Sullivan, and extended by the first author, Jones, Godin, and others.
We do this by studying maps from surfaces with cylindrical ends to M, such that
on the cylinders, they satisfy the gradient flow equation of a Morse function
on the loop space, LM. We then give Morse theoretic descriptions of related
constructions, such as the Thom and Euler classes of a vector bundle, as well
as the shriek, or unkehr homomorphism.Comment: 30 pages, 6 figures Final version to appear in Proc. of Conference on
Symplectic Field Theory in honor of the 60th birthday of Y. Eliashber
Stability for closed surfaces in a background space
In this paper we present a new proof of the homological stability of the
moduli space of closed surfaces in a simply connected background space ,
which we denote by . The homology stability of surfaces in with an
arbitrary number of boundary components, was studied by the
authors in \cite{cohenmadsen}. The study there relied on stability results for
the homology of mapping class groups, with certain families of
twisted coefficients. It turns out that these mapping class groups only have
homological stability when , the number of boundary components, is positive,
or in the closed case when the coefficient modules are trivial. Because of this
we present a new proof of the rational homological stability for , that
is homotopy theoretic in nature. We also take the opportunity to prove a new
stability theorem for closed surfaces in that have marked points.Comment: 14 page
Notes on string topology
This paper is an exposition of the new subject of String Topology. We present
an introduction to this exciting new area, as well as a survey of some of the
latest developments, and our views about future directions of research. We
begin with reviewing the seminal paper of Chas and Sullivan, which started
String Topology by introducing a BV-algebra structure on the homology of a loop
space of a manifold, then discuss the homotopy theoretic approach to String
Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat
graphs. We review quantum field theories and indicate how string topology fits
into the general picture. Other topics include an open-closed version of string
topology, a Morse theoretic interpretation, relation to Gromov-Witten
invariants, and "brane'' topology, which deals with sphere spaces. The paper is
a joint account of the lecture series given by each of us at the 2003 Summer
School on String Topology and Hochschild Homology in Almeria, Spain.Comment: 95 page
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