Metrics of positive scalar curvature and generalised Morse functions. II (Q2862122)

This is Part II of the author's PhD thesis [Metrics of positive scalar curvature and generalized Morse functions. Eugene, Oregon: University of Oregon (PhD Thesis) (2009)]. The thesis is a significant contribution to a larger project aimed at better understanding the topology of the space of metrics of positive scalar curvature (psc-metrics) on a given smooth manifold \(X\).NEWLINENEWLINEAs already mentioned in the review of Part I [Mem. Am. Math. Soc. 983, i-xvii, 80 p. (2011; Zbl 1251.53001)], a major goal of this project deals with connectedness properties in the space of psc-metrics \(\mathcal{R}\mathrm{iem}^+(X)\) on \(X\). Two metrics lying in the same path component of \(\mathcal{R}\mathrm{iem}^+(X)\) are called isotopic. Two psc-metrics \(g_0\) and \( g_1\) on \(X\) are called concordant if there exists a psc-metric \(\bar g\) on \(X\times[0,1]\) such that, near \(X\times\{0\}\), \(\bar g\) is the product \(g_0 +dt^ 2\), and near \( X\times\{1\}\), \(\bar g\) is the product \(g_1 +dt^2\).NEWLINENEWLINEArbitrary concordances can be very complicated. For a particular class of concordances, now called Gromov-Lawson concordances, \textit{M. Gromov} and \textit{H. B. Lawson jun.} [Ann. Math. (2) 111, 423--434 (1980; Zbl 0463.53025)] developed a surgery technique suitable for turning such concordances into isotopies.NEWLINENEWLINEIn Part I, the author used this technique to prove that Gromov-Lawson concordant metrics on closed simply connected manifolds of dimension at least 5 are isotopic, by modifying an admissible Morse function on the cobordism. In Part II the author extends this technique to work for families of generalised Morse functions, i.e., smooth functions that can have both Morse singularities and birth-death singularities. In a main result he proves that, under reasonable conditions, the isotopy type of a Gromov-Lawson cobordism does not depend on the choice of the Morse function and so is an invariant of the cobordism.