Twisted \(K\)-theory and obstructions against positive scalar curvature metrics (Q2877705)

Denote by NEWLINE\[NEWLINE\theta(M)=\mathrm{ind}(D^\nu_+) = [\nu^S] \otimes_{(C(M,\mathbb C\ell(M))} [D^S]NEWLINE\]NEWLINE the index of Dirac operator as the pairing of the twisted fundamental \(K\)-homology class \([D^S]\) of the Kasparov \(K\)-homology \(KK(C(M,\mathbb \ell(M), \mathbb C)\) and the twisted Mischenko-Fomenko bundle representing a twisted version of an element \([\nu^S]\) of the Kasparov \(K\)-theory \(KK(\mathbb C, \mathbb C\ell(M)) \otimes C^*(\hat{\pi} \to \pi))\), where \(\pi = \pi_1(M)\) is the fundamental group of \(M\) and \(\hat{\pi}\to \pi\) is the twisted fundamental group of \(M\) which is a \(\mathbb Z/(2)\) extension of \(\pi\). The main result of the paper is the theorem stating that for a closed smooth orientable even-dimensional manifold with dimension \(\dim(M) \geq 3\) and the universal cover \(\widetilde{M}\) which is spin and is enlargeable in the sense of Definition 4.1, then the obstruction \(\theta^{\max}(M)\) does not vanish.