Thom–Mather stratified space

In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof.

Basic examples of Thom–Mather stratified spaces include manifolds with boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex analytic varieties, or orbit spaces of smooth transformation groups.

Definition

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A Thom–Mather stratified space is a triple   where   is a topological space (often we require that it is locally compact, Hausdorff, and second countable),   is a decomposition of   into strata,

 

and   is the set of control data   where   is an open neighborhood of the stratum   (called the tubular neighborhood),   is a continuous retraction, and   is a continuous function. These data need to satisfy the following conditions.

  1. Each stratum   is a locally closed subset and the decomposition   is locally finite.
  2. The decomposition   satisfies the axiom of the frontier: if   and  , then  . This condition implies that there is a partial order among strata:   if and only if   and  .
  3. Each stratum   is a smooth manifold.
  4.  . So   can be viewed as the distance function from the stratum  .
  5. For each pair of strata  , the restriction   is a submersion.
  6. For each pair of strata  , there holds   and   (both over the common domain of both sides of the equation).

Examples

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One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety  , there is a naturally defined subvariety,  , which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus   will eventually give a natural stratification.[citation needed] A simple algebreo-geometric example is the singular hypersurface

      

where   is the prime spectrum.

See also

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References

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  • Goresky, Mark; MacPherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, 1988.
  • Goresky, Mark; MacPherson, Robert Intersection homology II, Invent. Math. 72 (1983), no. 1, 77--129.
  • Mather, J. Notes on topological stability, Harvard University, 1970.
  • Thom, R. Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
  • Weinberger, Shmuel (1994). The topological classification of stratified spaces. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 9780226885667.