Stratified Mappings — Structure and Triangulability

Verona, Andrei

doi:10.1007/bfb0101672

Cited by 84 publications

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“…To this end, take {/} a partition of unity made up of controlled functions (see [13]). It is easy to check that each function / is a liftable one (see [8, §4.1.5]).…”

Section: O^jç(b) -ï-* I3ç(m) -I-» Jq~x(b) -mentioning

confidence: 99%

Intersection cohomology of 𝑆¹-actions

Hector

Saralegi²

1993

Trans. Amer. Math. Soc.

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Abstract.Given a free action of the circle S1 on a differentiable manifold M , there exists a long exact sequence that relates the cohomology of M with the cohomology of the manifold M/S1 . This is the Gysin sequence. This result is still valid if we allow the action to have stationary points.In this paper we are concerned with actions where fixed points are allowed. Here the quotient space M/S1 is no longer a manifold but a stratified pseudomanifold (in terms of Goresky and MacPherson). We get a similar Gysin sequence where the cohomology of M/S1 is replaced by its intersection cohomology. As in the free case, the connecting homomorphism is given by the product with the Euler class [e]. Also, the vanishing of this class is related to the triviality of the action. In this Gysin sequence we observe the phenomenon of perversity shifting. This is due to the allowability degree of the Euler form.Given a free action

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“…To this end, take {/} a partition of unity made up of controlled functions (see [13]). It is easy to check that each function / is a liftable one (see [8, §4.1.5]).…”

Section: O^jç(b) -ï-* I3ç(m) -I-» Jq~x(b) -mentioning

confidence: 99%

Intersection cohomology of 𝑆¹-actions

Hector

Saralegi²

1993

Trans. Amer. Math. Soc.

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“…The "geometrically" stratified sets developed by R. Thorn [21] and others [22] have given an effective context for smooth and PL phenomena (for example real semialgebraic sets are smoothly stratified, and polyhedra are PL stratified), but the topological version has been less successful. Among other problems, there are examples which are very nice locally but which do not admit global geometrically stratified structures.…”

mentioning

confidence: 99%

Homotopically stratified sets

Quinn¹

1988

J. Amer. Math. Soc.

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The objective is to give a setting for the study of purely topological stratified phenomena, particularly group actions on manifolds. The "geometrically" stratified sets developed by R. Thorn [21] and others [22] have given an effective context for smooth and PL phenomena (for example real semialgebraic sets are smoothly stratified, and polyhedra are PL stratified), but the topological version has been less successful. Among other problems, there are examples which are very nice locally but which do not admit global geometrically stratified structures. As a result stratified topological questions have had to be approached in a somewhat ad hoc fashion.In a homotopically I stratified set the strata are related by homotopy rather than geometric conditions. This makes them easier to construct and far more general, but makes it harder to see that they have any useful properties. Nonetheless they do; the principal results of this paper give extensions of isotopies, collars for boundaries, recognition theorems, an h-cobordism theorem, and obstructions to the existence of regular neighborhoods and geometric stratifications. The isotopy extension theorem, for example, implies that the skeleta are topologically homogeneous.We give some background on stratifications, then discuss group actions, previous work, and the organization of the paper.A geometrically stratified set is a filtered space XO C Xl C ... C Xn such that the strata are manifolds. ("Skeleta" refers to the subsets Xi , "strata" the differences Xi -Xi-I.) The strata are also related to one another in a very precise way; if k > i then a neighborhood of Xi -Xi-I is a bundle of some kind, and there are compatibility conditions on these bundles at points where three or more strata come together. We get smooth, PL, or topological stratified sets depending on the types of strata and bundles used.In a homotopically stratified set the strata are related by homotopy conditions. Roughly, neighborhoods of Xi -Xi-I in (X k -X k -I ) U (Xi -Xi-I) have the local homotopy properties of mapping cylinders of fibrations. No compatibility conditions are required.

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“…The charts appearing in Definition 2.2 are C 1 -charts in which the Whitney-(b) condition holds. Therefore Σ is a Whitney-(b) stratification of X and Whitney stratified sets admit triangulations compatible with the stratification (see [14], [24], [46] and the references therein). By inspecting the proof of [14] and [15] one can check that in the case of an smcs the triangulation constructed in [14] is a smooth triangulation of the smooth manifold X.…”

Section: Submanifolds With Conical Singularities (Smcs) In the Sense mentioning

confidence: 99%

A complex in Morse theory computing intersection homology

Ludwig¹

2017

Annales De l'Institut Fourier

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Stratified Mappings — Structure and Triangulability

Cited by 84 publications

References 0 publications

Intersection cohomology of 𝑆¹-actions

Intersection cohomology of 𝑆¹-actions

Homotopically stratified sets

A complex in Morse theory computing intersection homology

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