nLab
transversal maps

Contents

Idea

Two maps f:XZf : X \to Z and g:YZg : Y \to Z of manifolds are transversal roughly if the images of XX and YY in ZZ do not “touch tangentially”.

Definition

Two maps f:XZf : X \to Z and g:YZg : Y \to Z of smooth manifolds are transversal if for all point xXx \in X and yYy \in Y with f(x)=z=g(y)f(x) = z = g(y) the differentials of ff and gg in these points span the entire tangent space at zz in the sense that

im(df)+im(dg)T zZ. im(d f) + im(d g) \simeq T_z Z \,.

Note that this is not required to be a direct sum. Also, if ff (say) is a submersion, then it is transversal to all gg.

In particular, ff or gg may be inclusions of (possibly immersed) submanifolds in which case we talk about the transversality of submanifolds.

Properties

Various constructions involving pullbacks of manifolds work as expected only for pullbacks involving transversal maps.

For example, two maps with a common target are transversal only if their pullback exists and is preserved by the tangent bundle functor; that is, T(X× ZY)=TX× TZTYT(X \times_Z Y) = T X \times_{T Z} T Y. (However, the pullback may exist and be preserved without transversality; for example if XX and YY are both abstract points, ZZ is not a point, and the maps X,YZX, Y \to Z are equal as concrete points of ZZ.)

This is to be regarded as the dual of the possibly more familiar statement that various constructions involving quotients only work as expected for free actions.

Both of these “problems” are solved by passing from the ordinary 11-category of manifolds to a suitable higher category of generalized smooth spaces.

More precisely:

References

  • T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; ∃ also 1990 German 2nd edition)

  • Morris W. Hirsch, Differential topology, Springer GTM 33, gBooks

Revised on June 5, 2013 23:31:24 by Toby Bartels (64.89.53.36)