Equivariant unfoldings of G-stratified pseudomanifolds

Abstract

The Intersection Cohomology with differential forms defined on stratified pseudomanifolds, introduced by Brasselet Hector and Saralegi; uses the unfolding given by Verona as an auxiliary tool. Broadly speaking, an unfolding over a stratified pseudomanifold X is a smooth manifold ̃X together with a continuous surjective map ℒₓ : ̃X → X such that the restriction to the regular part ℒₓ : ℒ¯¹(X − Σ) → X − Σ is a finite trivial covering, and for each singular stratum S the restriction ℒₓ : ℒ¯¹(S) → S is a smooth fiber bundle with fiber ̃LS ×ℝ where ̃Lₛ is the unfolding of the link Lₛ of the stratum S. The recursive method employed here turns this construction more difficult when the length of X is ≥ 1. In this work we introduce a class 𝔊 of G-transverse Thom-Mather stratified pseudomanifolds. An object of 𝔊 is an unfoldable stratified pseudomanifold X with arbitrary length, endowed with the action of a compact Lie group G such that the stratification of X is a refinement of the partition by orbit types (namely, the isotropy groups are con- stant over each stratum) and where the local conical structure is given by a slice of the action. Besides, we ask the stratification of X to be Thom-Mather compatible with the action, so there are equivariant tubular neighborhoods over the singular strata and these tubes are locally transverse to the action. A fundamental property is that for any object X ∈ 𝔊 and K ⊂ G a closed subgroup, the quotient space X/K is a G/K-transverse Thom-Mather stratified pseudomanifold, the quotient stratification being induced by the natural projection map. Since each smooth G-manifold is an object in 𝔊, this fact provides a rich source of examples of unfoldable stratified pseudomanifolds wich can be obtained starting on a smooth manifold