Some phenomena in tautological rings of manifolds

Randal-Williams, Oscar

doi:10.1007/s00029-018-0417-z

Cited by 5 publications

(2 citation statements)

References 30 publications

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“…The rings R .CP 2 /, R .CP 2 ; / and R .CP 2 ; D 4 / were investigated in [19], but their structure was not fully determined. By computing these rings we have settled some open problems posed in [19].…”

Section: Resultsmentioning

confidence: 99%

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Tautological classes of definite 4–manifolds

Baraglia

2023

Geom. Topol.

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ˆE=Bc.T .E=B// 2 H 4 .BI Q/;where ´E=B denotes integration over the fibres. Let Diff.X / denote the group of diffeomorphisms of X with the C 1 -topology (note that all diffeomorphisms of X are orientation-preserving since X is positive-definite) and BDiff.X / the classifying space. The tautological classes can be constructed for the universal bundle U X D EDiff.X / Diff.X / X, giving classesThe tautological ring of X, R .X / Â H .BDiff.X /I Q/;is defined as the subring of H .BDiff.X /I Q/ generated by tautological classes Ä c for c 2 H .BSO.4/I Q/. Since H .BSO.4/I Q/ is generated over Q by p 1 and e, it follows that R .X/ is generated by the classes fÄ p a 1 e b g a;b 0 . Similarly, for any family E ! B, we can define the tautological ring of E, R .E/ Â H .BI Q/; to be the subring generated by the tautological classes Ä c .E/ for c 2 H .BSO.4/I Q/.

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Section: Resultsmentioning

confidence: 99%

“…Combining the constant terms from (4-21) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and equating this to v ∅ , we obtain…”

Section: Proposition 42mentioning

confidence: 99%

Tautological classes of definite 4–manifolds

Baraglia

2023

Geom. Topol.

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Characteristic classes for families of bundles

Berglund

2022

Sel. Math. New Ser.

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The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying $$\tau _M$$ 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 $$\tau _M$$ τ M -fibrations, $$Baut(\tau _M)$$ B a u t ( τ M ) , and its cohomology ring, i.e., the ring of characteristic classes of $$\tau _M$$ τ M -fibrations. For a bundle $$\xi $$ ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal $$\xi $$ ξ -fibration with holonomy in a given connected monoid, 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 $$Baut(\xi )$$ B a u t ( 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 $$\tau _M$$ τ M -fibration comes from a manifold bundle.

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Tautological rings of fake quaternionic spaces

Prigge

2024

Sel. Math. New Ser.

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The tautological ring $$R^*(M)$$ R ∗ ( M ) of a smooth manifold M is the ring of characteristic classes generated by the Miller–Morita–Mumford classes, and is often more accessible than the ring of all characteristic classes of smooth M-bundles. In this paper, we introduce a new method to obtain upper bounds on the Krull dimension of $$R^*(M)$$ R ∗ ( M ) for manifolds homotopy equivalent to a fixed, simply connected Poincaré duality space by using recent progress in rational homotopy theory and the family signature theorem. In particular, we show that the Krull dimension of the tautological ring vanishes for almost all manifolds homotopy equivalent to $$\mathbb {H}P^2$$ H P 2 .

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Some phenomena in tautological rings of manifolds

Cited by 5 publications

References 30 publications

Tautological classes of definite 4–manifolds

Tautological classes of definite 4–manifolds

Characteristic classes for families of bundles

Tautological rings of fake quaternionic spaces

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