Abstract
In this thesis we introduce Δ-set  ^ψPL_d(R^N) which we regard as the piecewise linear analogue of the space ψ_d(R^N) of smooth d-dimensional submanifoldsin R^N introduced by Galatius in [4]. Using ψ^PL_d(R^N) we define a bi-Δ-set C_d(R^N)_•,• ( whose geometric realization BC^PL_d(R^N) = llC_d(R^N)_•,•ll should be interpreted as the PL version of the classifying space of the category of smooth d-dimensional cobordisms in R^N, studied in [7], and the main result of this thesis describes the weak homotopy type of BC^PL_d (R^N) in terms of ψ^PL_d (R^N)_•, namely, we prove that there is a weak homotopy equivalence BC^PL_d (R^N) ≅ Ω^N–1lψ^PL_d (R^N)_•l when N — d  ≥ 3.
The proof of the main theorem relies on properties of ψ^PL_d (R^N)• which arise from the fact that this Δ-set can be obtained from a more general contravariant functor PL^op → Sets defined on the category of finite dimensional polyhedraand piecewise linear maps, and on a fiberwise transversality result for piecewise linear submersions whose fibers are contained in R × (-1,1)^N-1 ⊆ R^N . For the proof of this transversality result we use a theorem of Hudson on extensions of piecewise linear isotopies which is why we need to include the condition N — d ≥ 3 in the statement of the main theorem.