Abstract
Abstract
In this thesis we introduce Δ-set ψPLd(RN) which we regard as the piecewise linear analogue of the space ψd(RN) of smooth d-dimensional submanifoldsin RN introduced by Galatius in [4]. Using ψPLd(RN) we define a bi-Δ-set Cd(RN)•,• ( whose geometric realization BCPLd(RN) = llCd(RN)•,•ll should be interpreted as the PL version of the classifying space of the category of smooth d-dimensional cobordisms in RN, studied in [7], and the main result of this thesis describes the weak homotopy type of BCPLd (RN) in terms of ψPLd (RN)•, namely, we prove that there is a weak homotopy equivalence BCPLd (RN) ≅ ΩN–1lψPLd (RN)•l when N — d ≥ 3.
The proof of the main theorem relies on properties of ψPLd (RN)• which arise from the fact that this Δ-set can be obtained from a more general contravariant functor PLop → 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 ⊆ RN . 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.
In this thesis we introduce Δ-set ψPLd(RN) which we regard as the piecewise linear analogue of the space ψd(RN) of smooth d-dimensional submanifoldsin RN introduced by Galatius in [4]. Using ψPLd(RN) we define a bi-Δ-set Cd(RN)•,• ( whose geometric realization BCPLd(RN) = llCd(RN)•,•ll should be interpreted as the PL version of the classifying space of the category of smooth d-dimensional cobordisms in RN, studied in [7], and the main result of this thesis describes the weak homotopy type of BCPLd (RN) in terms of ψPLd (RN)•, namely, we prove that there is a weak homotopy equivalence BCPLd (RN) ≅ ΩN–1lψPLd (RN)•l when N — d ≥ 3.
The proof of the main theorem relies on properties of ψPLd (RN)• which arise from the fact that this Δ-set can be obtained from a more general contravariant functor PLop → 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 ⊆ RN . 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.
| Original language | English |
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| Publisher | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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| ISBN (Print) | 978-87-7078-963-9 |
| Publication status | Published - 2014 |