A note on some classical results of Gromov-Lawson (Q2845882)

This paper uses higher index theory of spin Dirac operators to construct obstructions to existence of complete metrics with uniformly positive scalar curvature in the presence of enlargeable submanifolds. The index theoretical obstructions to positive scalar curvature metrics all arise from the Schrödinger-Lichnerowicz formula for the square of the spin Dirac operator. One of the main results of the paper is Theorem 3.2 stating that a spin manifold \(M\) does not admit a complete metric with uniformly positive scalar curvature outside a compact subset if there is an enlargeable closed hypersurface \(N\) partitioning the manifold into two non-compact components \(M_+\) and \(M_-\), and there is either a smooth mapping \(\overline{M_+}\to N\) having non-zero degree restricted to \(N\) or that the inclusion \(N\hookrightarrow \overline{M_+}\) induces an isomorphism on fundamental groups. Theorem 3.3 of the paper is a minor variation on Theorem 3.2 giving the same result if only \(M_+\), but not necessarily \(M_-\), is noncompact and \(N\) is area-enlargeable. Another important result in the paper is Theorem 3.4 which generalizes a result of Gromov-Lawson. Theorem 3.4 states that when \(N\) is an enlargeable closed submanifold of codimension \(2\) in a closed spin manifold \(M\), if \(\pi_1(N)\to \pi(M)\) is injective with infinite index then \(M\) does not admit a metric with positive scalar curvature.NEWLINENEWLINEThe first of the two main tools is a version of the partitioned manifold index theorem for coefficients in a flat \(C^*\)-bundle due to the author [J. Noncommut. Geom. 4, No. 3, 459--473 (2010; Zbl 1201.58020)]. The other main tool in the paper is Theorem 3.1 stating that an index invariant \(\mathrm{ind}(D,N)\), of the spin Dirac operator \(D\) relative to the compact hypersurface \(N\) of the complete Riemannian manifold \(M\), vanishes if the metric has uniformly positive scalar curvature outside a compact subset.NEWLINENEWLINEReviewer's remark: There are unfortunately some inaccuracies in the paper. In the proof of Theorem 3.1 a Friedrichs extension of a Hilbert module operator is used and claimed to give a self-adjoint extension of a Dirac operator on a manifold with boundary. Taking aside that it is not explained why the operator is closeable, the main inaccuracy in the proof of Theorem 3.1 is that the considered Friedrichs extension construction is of a Laplacian type operator and not the Dirac operator the author uses. Another inaccuracy is that there are missing assumptions in the statement of Theorem 3.4. As stated, Theorem 3.4 admits a counterexample. The counter example, and a sufficient set of assumptions for the theorem to hold were given in [\textit{B. Hanke, D. Pape} and \textit{T. Schick}, ``Codimension two index obstructions to positive scalar curvature'', \url{arXiv:1402.4094}].