Hyperbolic 4-manifolds with perfect circle-valued Morse functions

Battista, Ludovico; Martelli, Bruno

doi:10.1090/tran/8542

Hyperbolic 4-manifolds with perfect circle-valued Morse functions
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by Ludovico Battista and Bruno Martelli PDF

Trans. Amer. Math. Soc. 375 (2022), 2597-2625 Request permission

Abstract:

We exhibit some (compact and cusped) finite-volume hyperbolic $4$-manifolds $M$ with perfect circle-valued Morse functions, that is circle-valued Morse functions $f\colon M \to S^1$ with only index 2 critical points. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one.

An immediate consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds $M$ having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers $b_1(M), b_3(M)$ and rank $\mathrm {rk}(\pi _1(M))$.

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