Algebra/Topology seminar

Mauricio Gomez Lopez:

Spaces of piecewise linear manifolds

Abstract:

In their article 'Monoids of moduli spaces of manifolds' Galatius and Randal-Williams 
showed that the classifying space BC_d(R^n) of the d-dimensional cobordism category in R^n is weak homotopy equivalent to  \Omega^n-1 Th(\Gamma^\perp_n,d), the n-1 fold loop space of the Thom space of the vector bundle orthogonal to the tautological vector bundle over the grassmannian of d-planes in R^n.  They did this by introducing a Space of manifolds \Psi_d(R^n) and showing that  there are weak equivalences  BC_d(R^n) ~ \Omega^n-1 \Psi_d(R^n) and \Psi_d(R^n)~Th(\Gamma^\perp_n,d).  In this talk I will present the main result of my thesis, namely, a piecewise linear version of the first equivalence BC_d(R^n) ~ \Omega^n-1 \Psi_d(R^n). During this talk I will introduce the pl analogues of the spaces  BC_d(R^n) and \Psi_d(R^n) and indicate the steps of the proof of the equivalence BC_d(R^n) ~ \Omega^n-1 \Psi_d(R^n) in the piecewise linear setting.