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Reconstructing 4-manifolds from Morse 2-functions. (English) Zbl 1267.57020

Kirby, Rob (ed.) et al., Proceedings of the Freedman Fest. Based on the conference on low-dimensional manifolds and high-dimensional categories, Berkeley, CA, USA, June 6–10, 2011 and the Freedman symposium, Santa Barbara, CA, USA, April 15–17, 2011 dedicated to Mike Freedman on the occasion of his 60th birthday. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 18, 103-114 (2012).
Let \(X^n\) and \(\Sigma ^2\) be smooth, oriented, closed, connected manifolds. The authors [Proc. Natl. Acad. Sci. USA 108, No. 20, 8122–8125 (2011; Zbl 1256.57024)] have defined a Morse \(2\)-function \(f:X^n \rightarrow \Sigma ^2\) as a smooth function such that each point \(x_0 \in X^n\) has a coordinate chart \(\mathbb{R}\times \mathbb{R}^{n-1}\) and \(f(x_0)\) has a coordinate chart \(\mathbb{R}\times \mathbb{R}\) for which \(f(x)=f(t,y)=(t,f_t(y))\), where \(f_t : \mathbb{R}^{n-1}\rightarrow \mathbb{R}\) is a generic \(1\)-parameter family of smooth functions with non-degenerate critical points and birth and deaths. Under appropriate conditions on the homotopy class of the Morse \(2\)-function \(f\), in particular when \(\Sigma =S^2\), \(f\) is homotopic to a Morse \(2\)-function for which all fibers are connected and all critical points are indefinite, i.e., in local coordinates, \(f(t,y)=(t,-y_1^2 \pm y_2^2\pm \cdots \pm y_{n-2}^2 +y_{n-1}^2)\).
In this interesting paper, the authors give minimal conditions on the fold curves and fibers of a given Morse \(2\)-function \(f:X^4 \rightarrow S^2\), such that \(X^4\) and \(f\) can be reconstructed from a certain combinatorial diagram attached to \(S^2\) up to diffeomorphism (Theorem 1).
For the entire collection see [Zbl 1253.00022].

MSC:

57M50 General geometric structures on low-dimensional manifolds
57R45 Singularities of differentiable mappings in differential topology
57R65 Surgery and handlebodies
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R70 Critical points and critical submanifolds in differential topology

Citations:

Zbl 1256.57024
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