Wolfgang Steimle

The space of metrics of positive scalar curvature

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics.Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing

Splitting the relative assembly map, Nil-terms and involutions

\hat{A}

On the map of Bökstedt–Madsen from the cobordism category to A–theory

-genera, thus establishing the non-multiplicativity of the

Degeneracies in quasi-categories

\hat{A}

Parametrized cobordism categories and the Dwyer–Weiss–Williams index theorem

-genus in fiber bundles with simply connected base.

Approximately fibering a manifold over an aspherical one

We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.

Obstructions to stably fibering manifolds

Bokstedt and Madsen defined an infinite loop map from the embedded d‐dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic K‐ theory of BO.d/ in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it extends the universal parametrized A‐theory Euler characteristic of smooth bundles with compact d‐ dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map Q.BO.d/C/! A.BO.d//. 19D10, 55R12, 57R90

Obstructions to fibering a manifold

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent quasi-categories. The method is purely combinatorial and extends to semisimplicial objects in other categories; in particular to semi-simplicial spaces satisfying the Segal condition (semi-Segal spaces).

A twisted Bass–Heller–Swan decomposition for the algebraic K-theory of additive categories

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer–Weiss–Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.

Higher Whitehead torsion and the geometric assembly map

The paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers, i.e., is homotopic to manifold approximate fibration. We define obstructions in algebraic K-theory. Their vanishing is necessary and under certain conditions sufficient. Basic ingredients are Quinn’s thin h-cobordism theorem and end theorem, and knowledge about the Farrell–Jones conjectures in algebraic K- and L-theory and the MAF-rigidity conjecture by Hughes–Taylor–Williams.