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      Witten's perturbation on strata with general adapted metrics

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          Abstract

          Let \(M\) be a stratum of a compact stratified space \(A\). It is equipped with a general adapted metric \(g\), which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, \(g\) has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then \(g\) is called good. We consider the maximum/minimun ideal boundary condition, \(d_{\text{max/min}}\), of the compactly supported de~Rham complex on \(M\), in the sense of Br\"uning-Lesch, defining the cohomology \(H^*_{\text{max/min}}(M)\), and with corresponding Laplacian \(\Delta_{\text{max/min}}\). The first main theorem states that \(\Delta_{\text{max/min}}\) has a discrete spectrum satisfying a weak form of the Weyl's asymptotic formula. The second main theorem is a version of Morse inequalities using \(H_{\text{max/min}}^*(M)\) and what we call rel-Morse functions. The proofs of both theorems involve a version for \(d_{\text{max/min}}\) of the Witten's perturbation of the de~Rham complex, as well as certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on \(g\) to be good is general enough in the following sense: using intersection homology when \(A\) is a stratified pseudomanifold, for any perversity \(\bar p\le\bar m\), there is an associated good adapted metric on \(M\) satisfying the Nagase isomorphism \(H^r_{\text{max}}(M)\cong I^{\bar p}H_r(A)^*\) (\(r\in\mathbb{N}\)). If \(M\) is oriented and \(\bar p\ge\bar n\), we also get \(H^r_{\text{min}}(M)\cong I^{\bar p}H_r(A)\). Thus our version of the Morse inequalities can be described in terms of \(I^{\bar p}H_*(A)\).

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          Author and article information

          Journal
          2015-07-27
          2016-04-01
          Article
          1507.07589
          afe385a5-81a3-46e0-9731-8ee7291d1ddb

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          58A14, 58G11, 57R30
          48 pages
          math.DG math.AT math.GT

          Geometry & Topology
          Geometry & Topology

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