*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, ...
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This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.Less

This chapter deals with the stable parametrized *h*-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of *C*superscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic *K*-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter reduces the proof of the manifold part of the stable parametrized h-cobordism theorem to a result about spaces of stably framed manifolds. Here Δsuperscript q denotes the standard ...
More

This chapter reduces the proof of the manifold part of the stable parametrized h-cobordism theorem to a result about spaces of stably framed manifolds. Here Δsuperscript q denotes the standard affine q-simplex. All polyhedra will be compact, and all manifolds considered will be compact PL manifolds. The chapter begins with a discussion of spaces of PL manifolds. It defines a space of manifolds as a simplicial set, with families of manifolds parametrized by Δsuperscript q as the q-simplices. Relevant terms such as tangent microbundle, fiberwise tangent microbundle, stably framed family of manifolds, and space of stably framed n-manifolds are taken into account. The chapter also describes the spaces of thickenings and how to straighten the thickenings.Less

This chapter reduces the proof of the manifold part of the stable parametrized *h*-cobordism theorem to a result about spaces of stably framed manifolds. Here Δsuperscript *q* denotes the standard affine *q*-simplex. All polyhedra will be compact, and all manifolds considered will be compact PL manifolds. The chapter begins with a discussion of spaces of PL manifolds. It defines a space of manifolds as a simplicial set, with families of manifolds parametrized by Δsuperscript *q* as the *q*-simplices. Relevant terms such as tangent microbundle, fiberwise tangent microbundle, stably framed family of manifolds, and space of stably framed *n*-manifolds are taken into account. The chapter also describes the spaces of thickenings and how to straighten the thickenings.

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0004
- Subject:
- Mathematics, Geometry / Topology

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