Spaces of Piecewise Linear Manifolds
Publikation: Bog/antologi/afhandling/rapport › Ph.d.-afhandling › Forskning
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Spaces of Piecewise Linear Manifolds. / Gomez Lopez, Mauricio Esteban.
Department of Mathematical Sciences, Faculty of Science, University of Copenhagen, 2014.Publikation: Bog/antologi/afhandling/rapport › Ph.d.-afhandling › Forskning
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TY - BOOK
T1 - Spaces of Piecewise Linear Manifolds
AU - Gomez Lopez, Mauricio Esteban
PY - 2014
Y1 - 2014
N2 - AbstractIn 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.
AB - AbstractIn 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.
UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122036915905763
M3 - Ph.D. thesis
SN - 978-87-7078-963-9
BT - Spaces of Piecewise Linear Manifolds
PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
ER -
ID: 130762223