Orbit categories, classifying spaces, and generalized homotopy fixed points

We give a new description of Rosenthal's generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.Comment: 10 pages. Version 4: various improvements to the exposition based on comments from the referee and edito

How efficiently can one untangle a double-twist? Waving is believing!

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, geometrically defined untangling procedure, leading to new conclusions regarding the minimum possible complexity of untanglings. We animate and analyze how our untangling operates on frames in 3-space, and teach readers in a video how to wave the nullhomotopy with their hands.Comment: To appear in The Mathematical Intelligencer. For supplemental videos, see http://www.math.iupui.edu/~dramras/double-tip.html , or https://www.youtube.com/playlist?list=PLAfnEXvHU52ldJaOye-8kZV_C1CjxGx2C . For a supplemental virtual reality experience, see http://meglab.wikidot.com/visualizatio

The topological Atiyah-Segal map

Associated to each finite dimensional linear representation of a group $G$, there is a vector bundle over the classifying space $BG$. We introduce a framework for studying this construction in the context of infinite discrete groups, taking into account the topology of representation spaces. This involves studying the homotopy group completion of the topological monoid formed by all unitary (or general linear) representations of $G$, under the monoid operation given by block sum. In order to work effectively with this object, we prove a general result showing that for certain homotopy commutative topological monoids $M$, the homotopy groups of $\Omega BM$ can be described explicitly in terms of unbased homotopy classes of maps from spheres into $M$. Several applications are developed. We relate our constructions to the Novikov conjecture; we show that the space of flat unitary connections over the 3-dimensional Heisenberg manifold has extremely large homotopy groups; and for groups that satisfy Kazhdan's property (T) and admit a finite classifying space, we show that the reduced $K$-theory class associated to a spherical family of finite dimensional unitary representations is always torsion.Comment: 57 pages. Comments welcome

A finite dimensional approach to the strong Novikov conjecture

The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using the Atiyah-Singer families index theorem, and partly by Carlsson's deformation $K$--theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the $K$--theory and cohomology of representation spaces