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Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.

Roughly speaking, to any space M with perfect obstruc- tion theory we associate a space N with symmetric perfect obstruction theory. It is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (−1)-shifted cotangent bundle N. By localising from N to its C∗-fixed locus M this gives five notions of virtual signed Euler characteristic of M: (1) TheCiocan-Fontanine-Kapranov/Fantechi-G ̈ottschesignedvirtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande’s virtual Atiyah-Bott localisation of the virtual cycle of N to M, (3) Behrend’s Kai-weighted Euler characteristic localisation of the vir- tual cycle of N to M, (4) Kiem-Li’s cosection localisation of the virtual cycle of N to M, (5) (−1)vd times by the topological Euler characteristic of M. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.