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An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds. (English) Zbl 1377.58017
Summary: We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold \(N\) are obstructions to positive scalar curvature on an ambient manifold \(M\) that contains \(N\) as a submanifold. In the main result of this note, we show that the Rosenberg index of \(N\) is an obstruction to positive scalar curvature on \(M\) if \(N \hookrightarrow M \twoheadrightarrow B\) is a fiber bundle of spin manifolds with \(B\) aspherical and \(\pi_1(B)\) of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension-one submanifolds. We also discuss some elementary obstructions using the \(\hat{\operatorname{A}}\)-genus of certain submanifolds.

MSC:
58J22 Exotic index theories on manifolds
46L80 \(K\)-theory and operator algebras (including cyclic theory)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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References:
[1] 10.1007/s10711-005-9025-0 · Zbl 1095.53032
[2] 10.1515/CRELLE.2008.061 · Zbl 1154.46042
[3] 10.24033/asens.1361 · Zbl 0415.31001
[4] 10.5802/aif.3000 · Zbl 1344.58012
[5] 10.1007/s10711-008-9266-9 · Zbl 1149.19006
[6] 10.1017/S0305004100071425 · Zbl 0792.55001
[7] 10.1090/cbms/090
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