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2 results on '"57S15, 57S20, 58K20"'

1. Triangulation of the map of a $G$-manifold to its orbit space

Author

Murayama, Mitsutaka and Shiota, Masahiro

Subjects
Mathematics - Geometric Topology, 57S15, 57S20, 58K20
Abstract

Let $G$ be a Lie group and $M$ a smooth proper $G$-manifold. Let $pi:Mto M/G$ denote the natural map to the orbit space. Then there exist a PL manifold $P$, a polyhedron $L$ and homeomorphisms $tau:Pto M$ and $\sigma:M/Gto L$ such that $\sigma\circpi\circ\tau$ is PL. If $M$ and the $G$-action are of analytic class, we can choose subanalytic $\tau$ and then unique $P$ and $L$.

Published
2010
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2. TRIANGULATION OF THE MAP OF A G-MANIFOLD TO ITS ORBIT SPACE

Author

Masahiro Shiota and Mitsutaka Murayama

Subjects
Triangulation (topology), General Mathematics, Lie group, Geometric Topology (math.GT), 57S15, 57S20, 58K20, Space (mathematics), 58K20, Manifold, Piecewise linear function, Combinatorics, Mathematics::Logic, Polyhedron, Mathematics - Geometric Topology, 57S20, FOS: Mathematics, Orbit (control theory), 57S15, Mathematics
Abstract

Let G be a Lie group, and let M be a smooth proper G-manifold. Let M/G denote the orbit space, and let π : M → M/G be the natural map. It is known that M/G is homeomorphic to a polyhedron. In the present paper we show that there exist a piecewise linear (PL) manifold P, a polyhedron L, and homeomorphisms τ : P → M and σ : M/G → L such that σ o π o τ is PL. This is an application of the theory of subanalytic sets and subanalytic maps of Shiota. If M and the G-action are, moreover, subanalytic, then we can choose τ and σ subanalytic and P and L unique up to PL homeomorphisms.

Published
2013

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