Hyperbolic 4-manifolds with perfect circle-valued Morse functions. (English) Zbl 1490.57024
The study of circle-valued Morse functions was initiated by S. P. Novikov in 1980 while investigating a problem in hydrodynamics.
Given a compact \(n\)-dimensional manifold \(M\), possibly with boundary, a circle-valued Morse function on \(M\) is a smooth function with values in \(S^1\), \(f:M\rightarrow S^1\), such that all its critical points are non-degenerate and \(f|_{\partial M}\) has no critical points. A circle-valued Morse function \(f:M\rightarrow S^1\) is called perfect if the number of critical points of \(f\) is equal to \(|\chi(X)|\). In the odd-dimensional case, one has that \(\chi(M)=0\), hence a perfect circle-valued Morse function is a fibration. In the two-dimensional case, every closed orientable surface admits a perfect circle-valued Morse function.
In the paper under review, the authors investigate the existence of perfect circle-valued Morse functions on finite-volume hyperbolic \(4\)-manifolds which are compact or cusped. Note that a finite-volume hyperbolic \(4\)-manifold \(M\) is either closed or the interior of a compact manifold with boundary \(M^{\ast}\). Hence, in the latter case, one defines a circle-valued Morse function on \(M\) as the restriction of one on \(M^{\ast}\).
In this paper, the authors construct two cusped hyperbolic \(4\)-manifolds \(W\) and \(X\) and one closed hyperbolic \(4\)-manifold \(Z\), each of which admits a perfect circle-valued Morse function. These manifolds are built by coloring some right-angled polytopes. The circle-valued Morse functions on these manifolds are built using some techniques presented in [M. Bestvina and N. Brady, Invent. Math. 129, No. 3, 445–470 (1997; Zbl 0888.20021); M. Conder and C. Maclachlan, Proc. Am. Math. Soc. 133, No. 8, 2469–2476 (2005; Zbl 1071.57013)] together with some arguments presented in the paper.
Moreover, the authors establish some topological information about \(W\), \(X\), and \(Z\), such as the number of cusps, the Euler characteristic, and the Betti numbers over \(\mathbb{R}\). Furthermore, the authors prove that there are infinitely many finite-volume (compact and cusped) hyperbolic \(4\)-manifolds \(M\) with bounded Betti numbers \(\beta_1 (M)\) and \(\beta_3 (M)\) and rank of \(\pi_1 (M)\).
Given a compact \(n\)-dimensional manifold \(M\), possibly with boundary, a circle-valued Morse function on \(M\) is a smooth function with values in \(S^1\), \(f:M\rightarrow S^1\), such that all its critical points are non-degenerate and \(f|_{\partial M}\) has no critical points. A circle-valued Morse function \(f:M\rightarrow S^1\) is called perfect if the number of critical points of \(f\) is equal to \(|\chi(X)|\). In the odd-dimensional case, one has that \(\chi(M)=0\), hence a perfect circle-valued Morse function is a fibration. In the two-dimensional case, every closed orientable surface admits a perfect circle-valued Morse function.
In the paper under review, the authors investigate the existence of perfect circle-valued Morse functions on finite-volume hyperbolic \(4\)-manifolds which are compact or cusped. Note that a finite-volume hyperbolic \(4\)-manifold \(M\) is either closed or the interior of a compact manifold with boundary \(M^{\ast}\). Hence, in the latter case, one defines a circle-valued Morse function on \(M\) as the restriction of one on \(M^{\ast}\).
In this paper, the authors construct two cusped hyperbolic \(4\)-manifolds \(W\) and \(X\) and one closed hyperbolic \(4\)-manifold \(Z\), each of which admits a perfect circle-valued Morse function. These manifolds are built by coloring some right-angled polytopes. The circle-valued Morse functions on these manifolds are built using some techniques presented in [M. Bestvina and N. Brady, Invent. Math. 129, No. 3, 445–470 (1997; Zbl 0888.20021); M. Conder and C. Maclachlan, Proc. Am. Math. Soc. 133, No. 8, 2469–2476 (2005; Zbl 1071.57013)] together with some arguments presented in the paper.
Moreover, the authors establish some topological information about \(W\), \(X\), and \(Z\), such as the number of cusps, the Euler characteristic, and the Betti numbers over \(\mathbb{R}\). Furthermore, the authors prove that there are infinitely many finite-volume (compact and cusped) hyperbolic \(4\)-manifolds \(M\) with bounded Betti numbers \(\beta_1 (M)\) and \(\beta_3 (M)\) and rank of \(\pi_1 (M)\).
Reviewer: Dahisy Lima (Santo André)
MSC:
| 57K40 | General topology of 4-manifolds |
| 57R70 | Critical points and critical submanifolds in differential topology |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 57N16 | Geometric structures on manifolds of high or arbitrary dimension |
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