Transversality theorem: a useful tool for establishing genericity

AbstractWe present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney (a)-regularity of a stratification is necessary and sufficient for the stability of transversality.

Download Full-text

SOME BASIC RESULTS FOR PROPER FREE G-MANIFOLDS, WHERE G IS A DISCRETE GROUP

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500000559 ◽

2002 ◽

Vol 45 (1) ◽

pp. 43-48

Author(s):

Marja Kankaanrinta

Keyword(s):

Discrete Group ◽

Dense Subset ◽

Subject Classification ◽

Open Dense Subset ◽

Equivariant Maps ◽

Countable Discrete Group ◽

Transversality Theorem ◽

Equivariant Version

AbstractLet $G$ be a countable discrete group and let $M$ be a proper free $C^r$ $G$-manifold and $N$ a $C^r$ $G$-manifold, where $1\leq r\leq\omega$. We prove that if $G$ acts properly and freely also on $N$ and if $\dim(N)\geq2\dim(M)$, then equivariant immersions form an open dense subset in the space $C^r_G(M,N)$ of all equivariant $C^r$ maps from $M$ to $N$. The space $C^r_G(M,N)$ is equipped with a topology, which coincides with the Whitney $C^r$ topology if $G$ is finite and is suited to studying equivariant maps. We also prove an equivariant version of Thom’s transversality theorem and show that $C^\omega_G(M,N)$ is dense in $C^r_G(M,N)$, for $1\leq r\leq\infty$.AMS 2000 Mathematics subject classification: Primary 57S20

Download Full-text

Cobordism of homology manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100050507 ◽

1972 ◽

Vol 71 (2) ◽

pp. 247-270 ◽

Cited By ~ 3

Author(s):

N. Martin

Keyword(s):

Bundle Theory ◽

Transversality Theorem ◽

Cobordism Ring ◽

Homology Cobordism ◽

Homology Manifolds

It is the object of this paper to prove a transversality theorem for homology manifolds which is strong enough to be then able to prove a Thom theorem for the cobordism ring of homology manifolds. The new bundle theory necessary for this follows on naturally from that developed in ‘Homology Cobordism Bundles’(2).

Download Full-text