Indefinite Morse 2-functions: broken fibrations and generalizations. (English) Zbl 1328.57019

Gay, David T.; Kirby, Robion

Geom. Topol. 19, No. 5, 2465-2534 (2015).

A Morse 2-function is a generic smooth map from a manifold \(M^n\) to a surface, like classical Morse functions are generic smooth maps from \(M^n\) to \({\mathbb R}^1\). The singularities of 2-Morse functions are folds and cusps, where folds look locally like \((t, x_1,\ldots,x_{n-1})\to (t,f(x_1,\ldots,x_{n-1}))\) for a standard Morse singularity \(f\), and cusps look locally like \((t; x_1,\ldots,x_{n-1})\to(t, f_t(x_1,\ldots,x_{n-1}))\) for a standard birth \(f_t\) of a cancelling pair of Morse singularities.
The paper under review develops techniques for working with Morse 2-functions and generic homotopies between them, paying particular attention to avoiding definite folds, in which the modeling function \(f\) is a local extremum, and to guaranteeing connected fibers. When definite folds are avoided, the Morse 2-function (or generic homotopy) is said to be indefinite.
The main result is to extend the existence and uniqueness results to indefinite, Morse 2-functions with connected fibers. These results are proved for the general case that \(M^n\) is a manifold with boundary, mapping to a surface \(\Sigma^2\) with boundary. In the closed case the results specialize as follows.
For \(n>2\) a map \(f: M^n\to \Sigma^2\) from a closed \(n\)-manifold to a surface is homotopic to an indefinite Morse 2-function if and only if \(f_*(\pi_1M)\) has finite index in \(\pi_1\Sigma\). (The necessity of this condition is due to O. Saeki [Kyushu J. Math. 60, No. 2, 363–382 (2006; Zbl 1113.57016)]). If \(n>3\), then any such indefinite Morse 2-functions are homotopic through an indefinite generic homotopy.

Reviewer: Thilo Kuessner (Seoul)

MSC:

57M50	General geometric structures on low-dimensional manifolds
57R17	Symplectic and contact topology in high or arbitrary dimension

Keywords:

broken fibration; Morse function; Cerf theory; definite fold; elliptic umbilic

Citations:

Zbl 1113.57016

References:

[1]	S Akbulut, \cC Karakurt, Every \(4\)-manifold is BLF, J. Gökova Geom. Topol. 2 (2008) 83 · Zbl 1209.57015
[2]	D Asimov, Round handles and nonsingular Morse-Smale flows, Ann. of Math. 102 (1975) 41 · Zbl 0316.57020 · doi:10.2307/1970972
[3]	D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043 · Zbl 1077.53069 · doi:10.2140/gt.2005.9.1043
[4]	R I Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. 2008 (2008) · Zbl 1179.57033 · doi:10.1093/imrn/rnn101
[5]	R I Baykur, N Sunukjian, Round handles, logarithmic transforms and smooth \(4\)-manifolds, J. Topol. 6 (2013) 49 · Zbl 1297.57064 · doi:10.1112/jtopol/jts027
[6]	J Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudoisotopie, Inst. Hautes Études Sci. Publ. Math. (1970) 5 · Zbl 0213.25202 · doi:10.1007/BF02684687
[7]	D T Gay, R C Kirby, Constructing Lefschetz-type fibrations on four-manifolds, Geom. Topol. 11 (2007) 2075 · Zbl 1135.57009 · doi:10.2140/gt.2007.11.2075
[8]	D T Gay, R C Kirby, Fiber-connected, indefinite Morse \(2\)-functions on connected \(n\)-manifolds, Proc. Natl. Acad. Sci. USA 108 (2011) 8122 · Zbl 1256.57024 · doi:10.1073/pnas.1018465108
[9]	A Hatcher, J Wagoner, Pseudoisotopies of compact manifolds, Astérisque 6, Soc. Math. France (1973)
[10]	K Honda, Local properties of self-dual harmonic \(2\)-forms on a \(4\)-manifold, J. Reine Angew. Math. 577 (2004) 105 · Zbl 1065.53066 · doi:10.1515/crll.2004.2004.577.105
[11]	K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629 · Zbl 1071.58002 · doi:10.1216/rmjm/1181069872
[12]	R C Kirby, A calculus for framed links in \(S^3\), Invent. Math. 45 (1978) 35 · Zbl 0377.55001 · doi:10.1007/BF01406222
[13]	R C Kirby, P Melvin, P Teichner, Cohomotopy sets of \(4\)-manifolds (editors R Kirby, V Krushkal, Z Wang), Geom. Topol. Monogr. 18 (2012) 161 · Zbl 1273.57012 · doi:10.2140/gtm.2012.18.161
[14]	C LeBrun, Yamabe constants and the perturbed Seiberg-Witten equations, Comm. Anal. Geom. 5 (1997) 535 · Zbl 0901.53028
[15]	Y Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. 13 (2009) 277 · Zbl 1164.57006 · doi:10.2140/gt.2009.13.277
[16]	T Perutz, Lagrangian matching invariants for fibred four-manifolds, I, Geom. Topol. 11 (2007) 759 · Zbl 1143.53079 · doi:10.2140/gt.2007.11.759
[17]	T Perutz, Lagrangian matching invariants for fibred four-manifolds, II, Geom. Topol. 12 (2008) 1461 · Zbl 1144.53104 · doi:10.2140/gt.2008.12.1461
[18]	O Saeki, Elimination of definite fold, Kyushu J. Math. 60 (2006) 363 · Zbl 1113.57016 · doi:10.2206/kyushujm.60.363
[19]	C H Taubes, Seiberg-Witten and Gromov invariants for symplectic \(4\)-manifolds (editor R Wentworth), First Int. Press Lecture Series 2, International Press (2000) · Zbl 0967.57001
[20]	L R Taylor, The principal fibration sequence and the second cohomotopy set (editors R Kirby, V Krushkal, Z Wang), Geom. Topol. Monogr. 18 (2012) 235 · Zbl 1270.55012 · doi:10.2140/gtm.2012.18.235
[21]	M Usher, The Gromov invariant and the Donaldson-Smith standard surface count, Geom. Topol. 8 (2004) 565 · Zbl 1055.53064 · doi:10.2140/gt.2004.8.565
[22]	G Wassermann, Stability of unfoldings in space and time, Acta Math. 135 (1975) 57 · Zbl 0315.58010 · doi:10.1007/BF02392016
[23]	J Williams, The \(h\)-principle for broken Lefschetz fibrations, Geom. Topol. 14 (2010) 1015 · Zbl 1204.57027 · doi:10.2140/gt.2010.14.1015