Stably tangential strict hyperbolization

TL;DR

This paper advances hyperbolization techniques to preserve stable tangent bundles and constructs diverse hyperbolic manifolds with novel topological properties, including non-trivial characteristic classes.

Contribution

It introduces a hyperbolization method that maintains stable tangent bundles and produces hyperbolic manifolds with unique topological features.

Findings

Constructed hyperbolic manifolds with all non-top Stiefel--Whitney classes non-trivial.

Produced orientable hyperbolic manifolds with non-trivial Pontryagin classes.

Developed infinite towers of hyperbolic manifolds with no stably parallelizable covers.

Abstract

We show that the Charney--Davis strict hyperbolization procedure can preserve stable tangent bundles, answering a question of Charney and Davis. The key input is the construction of many hyperbolizing pieces, obtained using separability properties of hyperbolic cubulable groups. Moreover, these pieces may be chosen so that every face is connected, answering a question of Belegradek. We then apply this construction to suitable cubulations of flat manifolds to produce infinitely many commensurability classes of closed hyperbolic manifolds, both arithmetic and non-arithmetic, with diverse topological features. In particular, we obtain the first examples in which all the Stiefel--Whitney classes are non-trivial below the top degree, and the first orientable examples with non-trivial Pontryagin classes. We also construct infinite towers of finite covers of closed hyperbolic manifolds in which no cover is stably parallelizable or spin. Our methods further yield new pairs of exotic negatively curved Riemannian manifolds.