Piecewise-smooth manifold
In geometric topology, the piecewise-smooth or piecewise-differentiable manifolds are manifolds that have piecewise smooth transition maps between their local coordinate charts. These generalise the manifolds in DIFF (the category of smooth manifolds and smooth functions between them) and in PL (the category of piecewise linear manifolds and piecewise linear maps between them), allowing one to relate these properties. However, the piecewise-smooth maps are not closed under function composition, so the piecewise-smooth manifolds do not themselves form a category.
History
That every smooth (indeed, C1) manifold has a unique compatible PL structure (i.e. there exists a PL manifold, unique up to PL homeomorphism, that is piecewise-smoothly homeomorphic to the smooth manifold) was originally proven in (Whitehead 1940). A detailed expositionary proof is given in (Munkres 1966). The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in (Thurston 1997). A very brief outline is given in (McMullen 1997), while a short but detailed proof is given in (Lurie 2009).
References
- Lurie, Jacob (February 13, 2009), Whitehead Triangulations (Lecture 3) (PDF)
- McMullen, C. T. (21 Aug 1997). "Re: PL and DIFF manifolds: a question". Newsgroup: sci.math.research. Archived from the original on Apr 8, 2013. Retrieved May 10, 2012.
- Munkres, James R. (1966), Elementary Differential Topology, Annals of Mathematics Studies, vol. 54, Princeton University Press, ISBN 0-69109093-9, Chapter II
- Thurston, William (1997), Three-Dimensional Geometry and Topology, Princeton University Press, pp. 194–195, ISBN 978-0-69108304-9, PDIFF described as "piecewise smooth"
- Whitehead, J. H. C. (October 1940). "On C1-Complexes". The Annals of Mathematics. Second Series. 41 (4): 809–824. doi:10.2307/1968861. JSTOR 1968861.