Space of nonnegatively curved metrics and pseudoisotopies

Belegradek, Igor; Farrell, F. T.; Kapovitch, Vitali

doi:10.4310/jdg/1488503001

Cited by 9 publications

(16 citation statements)

References 30 publications

(40 reference statements)

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“…There has been much activity and profound progress on these issues in the last decades, compare, for example, . Since the very beginning, special importance has been given to the study of connectedness properties of spaces and moduli spaces of metrics with lower curvature bounds on closed as well as open manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

Dessai

Klaus

Tuschmann

2017

Bull. London Math. Soc.

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We show that in each dimension 4n + 3, n ≥ 1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

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Section: Introductionmentioning

confidence: 99%

“…The study of moduli spaces of nonnegative sectional curvature metrics on open manifolds was initiated in the paper , with follow‐ups in particular in the works .…”

Section: Introductionmentioning

confidence: 99%

Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

Dessai

Klaus

Tuschmann

2017

Bull. London Math. Soc.

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“…Especially in recent years there has been intensive activity and substantial further progress on these issues, compare, for example, [2,3,[6][7][8][9][10][11][13][14][15][16][17][18][20][21][22][25][26][27][28][29][30][31][33][34][35][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]58,60,61,[65][66][67][69]…”

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confidence: 99%

On the topology of moduli spaces of non-negatively curved Riemannian metrics

Tuschmann

Wiemeler

2021

Math. Ann.

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We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

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“…Belegradek, Farrell and Kapovitch proved in [BFK15] that for many open manifolds V admitting complete nonnegatively curved Riemannian metrics and for certain positive integers i the rational homotopy groups…”

Section: Application To the Space Of The Nonnegatively Curved Metricsmentioning

confidence: 99%

“…For example, when d ≥ 2 and for some explicit i ≥ 2, they prove that there exits an m such that π Q i R K≥0 (T S 2d × S m ) = 0. However, the methods in [BFK15] are insufficient to determine m exactly when i is given. It turns out, though, that this can be fixed by computing the ranks of Inv + π Q k P(M ) and Inv − π Q k P(M ) when M is the unit sphere bundle of S 2d .…”

Section: Application To the Space Of The Nonnegatively Curved Metricsmentioning

confidence: 99%

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

Bustamante

Farrell

Jiang

2020

Trans. Amer. Math. Soc.

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We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic K-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational S 1 -homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds V that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on V have nontrivial rational homotopy groups. * The third author's research is partially supported NSFC 11571343.

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Space of nonnegatively curved metrics and pseudoisotopies

Cited by 9 publications

References 30 publications

Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

On the topology of moduli spaces of non-negatively curved Riemannian metrics

Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

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