Spaces of Piecewise Linear Manifolds

In this talk I will present the results of my PhD thesis. I will start by introducing a Delta-set Psi_d(R^N) which we regard as the piecewise linear analogue of the space of smooth manifolds introduced by Galatius in the article 'Stable homology of automorphism groups of free groups'. Using Psi_d(R^N) we then define a bi-Delta-set whose geometric realization BC_d(R^N) shoud be interpreted as the PL version of the classifying space of the category of d-dimensional cobordisms embedded in N-dimensional Euclidean space, studied by Galatius and Randal-Williams in the article 'Monoids of moduli spaces of manifolds'. Finally, I will present the main result of my thesis, namely, the space BC_d(R^N) is weak homotopy equivalent to the (N-1)-fold loop space of Psi_d(R^N) when N-d>2.

 The proof of the main theorem relies on properties of Psi_d(R^N) which arise from the fact that this Delta-set can be obtained from a more general contravariant set valued functor defined on the category of finite dimensional polyhedra and piecewise linear maps, and on a fiberwise transversality result for piecewise linear submersions whose fibers are contained in the open subspace R x (-1,1)^{N-1} of R^N. For the proof of this transversality result I use a theorem of Hudson on extensions of piecewise linear isotopies which is why we need to include the condition N-d>2 in the statement of the main theorem.

Advisor: Prof. Erik Kjær Pedersen, Math, University of Copenhagen

Assessment committee:

Prof. Ib Madsen (chairman), MATH, University of Copenhagen

Prof. Søren Galatius, Stanford University, United States

Prof. Matthias Kreck, University of Bonn, Germany