Tautological classes and smooth bundles over BSU(2)

Abstract

For a Lie group G and a smooth manifold W , we study the difference between smooth actions of G on W and fiber bundles over the classifying space of G with fiber W and structure group Diff(W ). In particular, we exhibit smooth manifold bundles over BSU (2) that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes of the fiber bundle associated to a group action.

Parametrised moduli spaces of surfaces as infinite loop spaces

“…In the case 𝐺 = Z and 𝑌 = 𝑆 1 , there is a continuous functor Cob Z 𝑑 → Cob 𝑑 (𝑆 1 ), given by taking mapping tori: using that every smooth bundle over 𝑆 1 is induced from a diffeomorphism, this functor is in fact a levelwise equivalence. For more general groups G and 𝑌 = 𝐵𝐺, the analogous argument pertaining to G-actions and bundles over 𝐵𝐺 can fail: see [22] for a discussion of this phenomenon and counterexamples in case 𝐺 = SU (2).…”

“…It is natural to study two related generalisations of , in an equivariant and a parametrised direction: for a (topological) group G , we can consider the G -equivariant cobordism category : objects and morphisms are, respectively, - and d -manifolds endowed with an (continuous) action of G by orientation-preserving diffeomorphisms; for a topological space Y , we can consider the Y -parametrised cobordism category : objects and morphisms are, respectively, orientable - and d -manifold bundles over Y . In the case and , there is a continuous functor , given by taking mapping tori: using that every smooth bundle over is induced from a diffeomorphism, this functor is in fact a levelwise equivalence. For more general groups G and , the analogous argument pertaining to G -actions and bundles over can fail: see [22] for a discussion of this phenomenon and counterexamples in case .…”

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

“…For instance, in [MT01] the authors apply the theorem towards early progress on the Mumford conjecture. Similar theorems are used in [RW08] to compute the Z/2 homology of the stable nonorientable mapping class group, as well as in [Rei19] to establish the existence of non-kinetic smooth bundles over the classifying space BSU (2). Douglas [Dou06] gave alternative proofs to [BM76] using Dold's Euclidean neighborhood rectracts.…”

Parametrised moduli spaces of surfaces as infinite loop spaces

“…In the case G = Z and Y = S 1 there is a continuous functor Cob Z d → Cob d (S 1 ), given by taking mapping tori: using that every smooth bundle over S 1 is induced from a diffeomorphism, this functor is in fact a levelwise equivalence. For more general groups G and Y = BG, the analogous argument pertaining to G-actions and bundles over BG can fail: see [Rei19b] for a discussion of this phenomenon and counterexamples in case G = SU(2).…”