Scalar Curvature Estimates by Parallel Alternating Torsion

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500Comment: 17 page

Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.Comment: LaTeX, 40 pages; v2: references added, BL = IK announced; v3: reference to DWW = IK adde

An analytic invariant of G_2 manifolds

The first and third authors have constructed a defect invariant $\nu(M,\phi)$ in Z/48 for $G_2$-structures on a closed 7-manifold. We describe the nu-invariant using $\eta$-invariants and Mathai-Quillen currents on M and show that it can be refined to an integer-valued invariant $\bar\nu(M,g)$ for $G_2$-holonomy metrics. As an example, we determine the $\bar\nu$-invariants of twisted and extra twisted connected sums a la Kovalev, Corti-Haskins-Nordstr\"om-Pacini, and Nordstr\"om. In particular, we exhibit examples of 7-manifolds where the moduli space of $G_2$-holonomy metrics has at least two connected components. In one of these examples, the underlying $G_2$-structures are homotopic, in another one, they are not.Comment: pdfLaTeX, 26 pages, 2 figures (tikz). v2: 28 pages, 2 figures. Reorganised the contents for clarity. Comments welcome

The Extended Bloch Group and the Cheeger-Chern-Simons Class

We present a formula for the full Cheeger-Chern-Simons class of the tautological flat complex vector bundle of rank two over BSL(2,\C^\delta). Our formula improves the formula by Dupont and Zickert, where the class is only computed modulo 2-torsion.Comment: LaTeX2e, 9 page

Vafa-Witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.Comment: LaTeX, 11 pages. V2: Rigidity statement added, minor changes. To appea