nLab piecewise-linear manifold

Idea

A piecewise-linear manifold, or PL manifold for short, is a manifold whose transition functions are piecewise linear functions?.

Related concepts

• triangulation theorem

References

See also

• Wikipedia, Piecewise linear manifold

Branched covers:

Every $n$-dimensional PL manifold is a branched cover of the n-sphere:

• James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920) 370–373

All PL 4-manifolds are simple branched covers of the 4-sphere:

• Riccardo Piergallini, Four-manifolds as 4-fold branched covers of $S^4$, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, pdf)

• Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)

The n-torus for $n \geq 3$ is not a cyclic branched cover of the n-sphere:

• U. Hirsch, W. D. Neumann, On cyclic branched coverings of spheres, Math. Ann. (1975) 215: 289 (doi:10.1007/BF01343895)