Weight decompositions of Thom spaces of vector bundles in rational homotopy

Buijs, Urtzi; Morán, Federico Cantero; Cirici, Joana

doi:10.1007/s40062-019-00243-2

Cited by 4 publications

(4 citation statements)

References 26 publications

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“…The reduction is exactly the same as before but requires a generalization of Proposition 3.9: Proposition 7.6 [19,Thm. B]; see also the slightly weaker [7,Thm. 3.4].…”

Section: Upper Bounds On Degreementioning

confidence: 98%

“…To see how the latter situation arises, consider the simplest example of a nonformal simply connected manifold, given in [13, p. 94]. This is the total space M of a fiber bundle 𝑆 3 → 𝑀 → 𝑆 2 × 𝑆 2 obtained by pulling back the Hopf fibration 𝑆 3 → 𝑆 7 → 𝑆 4 along the degree 1 map 𝑆 2 × 𝑆 2 → 𝑆 4 .…”

Section: Theorem C Let Y Be a Formal Simply Connected Compact Riemann...mentioning

confidence: 99%

“…An L-Lipschitz self-map takes the generators of 𝐻 5 (𝑀) to vectors of length 𝑂 (𝐿 5 ), and therefore, it takes the generators of 𝐻 2 (𝑀) to vectors of length 𝑂 (𝐿 5/3 ). This means the degree of such a map is 𝑂 (𝐿 20/3 ) ≺ 𝐿 7 .…”

Section: Theorem C Let Y Be a Formal Simply Connected Compact Riemann...mentioning

confidence: 99%

“…(3) 22 ) | 𝑑𝑎 𝑖 = 0, 𝑑𝑏 𝑖 𝑗 = 𝑎 𝑖 𝑎 𝑗 and therefore, for any t, it has an automorphism which takes 𝑎 𝑖 ↦ → 𝑡𝑎 𝑖 and 𝑏 𝑖 𝑗 ↦ → 𝑡 2 𝑏 𝑖 𝑗 . Now,𝐻 5 (𝑀; Q) 𝑏 11 𝑎 2 − 𝑎 1 𝑏 12 , 𝑏 12 𝑎 2 − 𝑎 1 𝑏 22 𝐻 7 (𝑀; Q) 𝑏 11 𝑎 2 2 − 𝑎 1 𝑎 2 𝑏 12 ∼ 𝑎 2 1 𝑏 22 − 𝑎 1 𝑎 2 𝑏 12 ,and therefore, this automorphism multiplies elements of 𝐻5 (𝑀; Q) by 𝑡3 and elements of 𝐻7 (𝑀; Q) by 𝑡4 .A priori, automorphisms of the minimal model need not be realized by genuine maps of finite complexes. But manifolds with positive weights have self-maps of arbitrarily high degree[8, Theorem Now, suppose 𝑓 : 𝑀 → 𝑀 is of degree d and 𝑓 * : 𝐻 𝑘 (𝑀; C) → 𝐻 𝑘 (𝑀; C) has some eigenvalue 𝜆 such that |𝜆| ≠ 𝑑 𝑘/𝑛 .…”

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confidence: 99%

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Degrees of maps and multiscale geometry

Berdnikov,

Guth,

Manin

2024

Forum of Mathematics, Pi

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We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of k copies of $\mathbb CP^2$ for $k \ge 4$ , then we prove that the maximum degree of an L-Lipschitz self-map of $X_k$ is between $C_1 L^4 (\log L)^{-4}$ and $C_2 L^4 (\log L)^{-1/2}$ . More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $\sim L^n$ . For formal but nonscalable simply connected n-manifolds, the maximal degree grows roughly like $L^n (\log L)^{-\theta (1)}$ . And for nonformal simply connected n-manifolds, the maximal degree is bounded by $L^\alpha $ for some $\alpha < n$ .

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“…The reduction is exactly the same as before but requires a generalization of Proposition 3.9: Proposition 7.6 [19,Thm. B]; see also the slightly weaker [7,Thm. 3.4].…”

Section: Upper Bounds On Degreementioning

confidence: 98%