All Questions
Tagged with loop-spaces dg.differential-geometry
12 questions
7
votes
2
answers
500
views
Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
- 489
4
votes
0
answers
216
views
Exterior derivative on loop space
Notations:
Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of ...
- 562
6
votes
0
answers
253
views
The tangent space to the Hilbert manifold of $H^1$ loops at non-smooth loops
It is satisfactory to have a nice functional analytic setting for the energy functional in Riemannian geometry. I'm currently deep into Klingenberg's book "Riemannian geometry" which (among other ...
- 7,583
3
votes
0
answers
120
views
Differentiability of a map to the free loop space
While reading Morse theory, closed geodesics, and the homology
of free loop spaces, the author claims the following:
Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...
- 401
21
votes
1
answer
1k
views
Reconciling the affine grassmannian and the based loop group
I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...
- 627
4
votes
0
answers
151
views
Is this groupoid a model for the derived fixed-point locus of the free loop space?
In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...
- 35.5k
11
votes
1
answer
871
views
Vector fields on path spaces
I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable ...
- 2,035
13
votes
1
answer
731
views
free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...
- 2,035
7
votes
1
answer
1k
views
Haar measure on infinite dimensional Lie groups?
Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like ...
- 548
7
votes
4
answers
973
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...
- 5,063
3
votes
0
answers
267
views
Maps of loop spaces with infinity-bounded differential.
I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:
In the following piece-wise smooth means smooth on ...
- 2,590
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k