The differential topology of loop spaces

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category o...

We study the topology of the complex points of the algebraic loop space of a smooth curve.

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".

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 $\xi$. This is essentially a decomposition of $\xi$ as $\zeta \otimes L\mathbb C$, where $\zeta$ is a finite dimensional subbundle of $\xi$ and $L\mathbb C$ is the loop space of the complex numbers. The criterion is a reduction of the structure group to the finite ra...

We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these ta...

This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general overview are advised to start there. We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense ...

This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here we prove T-duality and Hori formulae for winding q-loop spaces, which are infinite dimensional subspaces of loop spaces.

We show that there is a topology on the group of loops in euclidean space such that this group is embedded in a Lie group which is simple relative to the loops. An extension of this Lie group gives the structural group of a principal bundle with connection whose holonomy coincides with the Chen signature map. We also give an alternative geometric new proof of Chen signature Theorem.

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.

Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.