A vanishing theorem for tautological classes of aspherical manifolds

Hebestreit, Fabian; Land, Markus; Lück, Wolfgang; Randal-Williams, Oscar

doi:10.48550/arxiv.1705.06232

Cited by 5 publications

(6 citation statements)

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“…e f w (π) = c f w n (π). We remark that e f w (π) as defined here agrees with the fiberwise Euler class in the sense of [18], as follows from [32,Theorem 5.6]. Let Baut(τ R CP n ) p,e denote the classifying space for τ R CP n -fibrations (π, ζ) with trivializations of the Pontryagin differences and the Euler difference,…”

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confidence: 75%

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Characteristic classes for families of bundles

Berglund¹

2020

Preprint

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The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber M depend only on the underlying τ M -fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for τ M -fibrations, Baut(τ M ), and its cohomology ring, i.e., the ring of characteristic classes of τ M -fibrations.For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable ξ-fibration together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(ξ) as well as the subring generated by the generalized Miller-Morita-Mumford classes.To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given τ M -fibration comes from a manifold bundle. Contents 24 4.1. Even spheres 24 4.2. Odd spheres 25 4.3. Complex projective spaces 27 References 41

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confidence: 75%

“…This yields the formula(19). The formula(21) is derived in a similar fashion by inserting e(L) = ω f w (π) − e |0 (π, L) in(18) and by observing that(19) in particular shows that c 1 (E) = (n + 1)e |0 (π, L).…”

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confidence: 92%

Characteristic classes for families of bundles

Berglund¹

2020

Preprint

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“…For manifolds of higher dimension they have recently been studied by Grigoriev, Galatius, and the author [Gri17, GGRW17, RW18]. In a different direction the vanishing of tautological classes for various aspherical manifolds has been shown by Bustamante, Farrell, and Jiang [BFJ16], and by Hebestreit, Land, Lück, and the author [HLLRW17]. A variant of tautological rings for Poincaré complexes rather than manifolds has been studied by Prigge [Pri19].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Tautological rings and stabilisation

Randal-Williams

2020

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“…Proof. This is proved in Section 3.1 of [HLLRW17] in dual form, where, passing to rational coefficients, the equivalence is expressed as…”

Section: Outlinementioning

confidence: 91%

Some phenomena in tautological rings of manifolds

Randal-Williams¹

2018

Sel. Math. New Ser.

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We prove several basic ring-theoretic results about tautological rings of manifolds W , that is, the rings of generalised Miller-Morita-Mumford classes for fibre bundles with fibre W . Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitelygenerated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W . Lastly, we give extensive computations in the tautological rings of CP 2 and S 2 × S 2 . arXiv:1611.03038v2 [math.AT] 6 May 2018 1.3. Krull dimension. Our second main result is a general technique, continuing on from our work with Galatius and Grigoriev [GGRW17, §4], for estimating the Krull dimension (for which we write Kdim) of the rings R * (W ) from below in terms of torus actions on W . The general statement is Theorem 3.1, but the hypotheses of that theorem are somewhat involved: we state here one of its corollaries with hypotheses which are easy to verify. Corollary B. Let a k-torus T act effectively on W , and suppose that either

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A vanishing theorem for tautological classes of aspherical manifolds

Cited by 5 publications

References 25 publications

Characteristic classes for families of bundles

Characteristic classes for families of bundles

Tautological rings and stabilisation

Some phenomena in tautological rings of manifolds

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