A refinement of Betti numbers and homology in the presence of a continuous function. I (Q2398912)

This paper describes how, given a continuous real-valued function on a compact ANR \(X\), one can get a refinement of the Betti numbers and homology of \(X\). The refinement creates a configuration of points in complex space, whose total cardinalities are the Betti numbers. The refinement of the homology consists of a set of vector spaces indexed by points in the complex plane with the same support as that of the point configuration. The direct sum of these vector spaces is the homology of \(X\). The author calls the system \((H_{r}(X), P_{r}^{f}(z), \hat{\delta}_{r}^{f})\), where \(P_{r}^{f}\) is a polynomial with zeros at the configuration points and \(\hat{\delta}_{r}^{f}\) an associated vector space, the \(r\)-homology spectral package of \((X,f)\). The points \((a,b)\) in the configuration arise by considering boxes of the form \((a', a] \times [b,b')\) and comparing the homology of the sub levels \(X_{a} = f^{-1}((-\infty , a])\) for \(a\) and \(a'\) and the homology of the super levels \(X^{b} = f^{-1}([b,+\infty))\) for \(b\) and \(b'\) to isolate the points where the homology of the levels changes. The author proves results on the stability of these spectral packages under changes in the function \(f\). A Poincaré duality result is shown for the case where \(X\) is a closed smooth manifold.