nLab equivariant triangulation theorem

Idea

The equivariant triangulation theorem (Illman 78, Illman 83) says that for $G$ a compact Lie group (for instance a finite group) and $X$ a compact smooth manifold equipped with a smooth $G$-action, there exists a $G$-equivariant triangulation of $X$.

(Illman 72, Thm. 3.1, Illman 83, theorem 7.1, corollary 7.2) Recalled as (ALR 07, theorem 3.2). See also Waner 80, p. 6 who attributes this to Matumoto 71

Moreover, if the manifold does have a boundary, then its G-CW complex may be chosen such that the boundary is a G-subcomplex. (Illman 83, last sentence above theorem 7.1)

These results continue to hold when $G$ is not compact, see Illman00.

Applications

• G-representation spheres are G-CW-complexes

Related concepts

• triangulation theorem

• equivariant good open cover

• equivariant differential topology

References

• Sören Illman, Theorem 3.1 in: Equivariant algebraic topology, Princeton University 1972 (pdf)

• Sören Illman, Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group, Math. Ann. 233 (1978) 199-220 [doi:10.1007/BF01405351]

• Sören Illman, The Equivariant Triangulation Theorem for Actions of Compact Lie Groups, Mathematische Annalen 262 (1983) 487-502 [dml:163720]

• Sören Illman, Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000) 129–183. [doi:10.1515/crll.2000.054]

See also

• Takao Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf, pdf)

• Stefan Waner, Equivariant Homotopy Theory and Milnor’s Theorem, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (JSTOR)

• A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (pdf)