The Joker is an important finite cyclic module over the mod-2 Steenrod algebra A. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory (2-compact groups, topological modular forms) and may be of independent interest.
We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces.
We compute the homology of the space of equivariant loops on the classifying space of a simplicial monoid M with anti-involution, provided pi(0)(M) is central in the homology ring of M. The proof is similar to McDuff and Segal's proof of the group completion theorem. Then we give an analogous computation of the homology of the C-2 -fixed points of a Gamma-space-type delooping of an additive category with duality with respect to the sign circle. As an application we show that this fixed-point space is sometimes group complete, but in general not.
Building on work of Livernet and Richter, we prove that En -homology and En - cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore, we show that the associated Yoneda algebra is trivial.