Mathematics

We explain the existence of a smooth H P 2 -bundle over S 4 whose total space has nontrivial A ^ -genus. Combined with an argument going back to Hitchin, this answers a question of Schick and implies that the space of Riemannian metrics of positive sectional curvature on a closed manifold can have nontrivial higher rational homotopy groups.

We consider a nontrivial action of C 2 on the type 1 spectrum Y:= M 2 (1)∧C(η) , which is well-known for admitting a 1 -periodic v 1 − self-map. The resultant finite C 2 -equivariant spectrum Y C 2 can also be viewed as the complex points of a finite R -motivic spectrum Y R . In this paper, we show that one of the 1 -periodic v 1 − self-maps of Y can be lifted to a self-map of Y C 2 as well as Y R . Further, the cofiber of the self-map of Y R is a realization of the subalgebra A R (1) of the R -motivic Steenrod algebra. We also show that the C 2 -equivariant self-map is nilpotent on the geometric fixed-points of Y C 2 .

Until now only for special classes of infra-solvmanifolds, namely infra-nilmanifolds and infra-solvmanifolds of type (R), there was a formula available for computing the Nielsen number of a self-map on those manifolds. In this paper, we provide a general averaging formula which works for all self-maps on all possible infra-solvmanifolds and which reduces to the old formulas in the case of infra-nilmanifolds or infra-solvmanifolds of type (R). Moreover, when viewing an infra-solvmanifold as a polynomial manifold, we recall that any map is homotopic to a polynomial map and we show how our formula can be translated in terms of the Jacobian of that polynomial map.

We show that an ∞ -category M left tensored over a monoidal ∞ -category V is completely determined by its graph M ≃ × M ≃ →P(V), (X,Y)↦M((−)⊗X,Y), parametrized by the maximal subspace M ≃ in M , equipped with the structure of an enrichment in the sense of Gepner-Haugseng in the Day-convolution monoidal structure on the ∞ -category P(V) of presheaves on V. Precisely, we prove that sending an ∞ -category left tensored over V to its graph defines an equivalence between ∞ -categories left tensored over V and a subcategory of all ∞ -categories enriched in presheaves on V. More generally we consider a generalization of ∞ -categories left tensored over V , which Lurie calls ∞ -categories pseudo-enriched in V , and extend the former equivalence to an equivalence χ between ∞ -categories pseudo-enriched in V and all ∞ -categories enriched in presheaves on V. The equivalence χ identifies V -enriched ∞ -categories in the sense of Lurie with V -enriched ∞ -categories in the sense of Gepner-Haugseng. Moreover if V is symmetric monoidal, we prove that sending an ∞ -category left tensored over V to its graph is lax symmetric monoidal with respect to the relative tensorproduct on ∞ -categories left tensored over V and the canonical tensorproduct on P(V) -enriched ∞ -categories.

We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by F 2 -synthetic spectra.

Let A be a small abelian category. The purpose of this paper is to introduce and study a category A ¯ ¯ ¯ ¯ which implicitly appears in construction of some TQFTs where A ¯ ¯ ¯ ¯ is determined by A . If A is the category of abelian groups, then the TQFTs obtained by Dijkgraaf-Witten theory of abelian groups or Turaev-Viro theory of bicommutative Hopf algebras factor through A ¯ ¯ ¯ ¯ up to a scaling. In this paper, we go further by giving a sufficient condition for an A -valued Brown functor to extend to a homotopy-theoretic analogue of A ¯ ¯ ¯ ¯ -valued TQFT for arbitrary A . The results of this paper and our subsequent paper reproduces TQFTs obtained by DW theory and TV theory.

We show that normalized cacti form an ∞ -operad in the form of a dendroidal space satisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketed trees and an enrichment of the dendroidal category Ω .

In this article we discuss Bousfield localization, beginning with definitions in terms of mapping spaces and working up to a discussion of how they can be constructed when we have access to the small object argument. We also discuss Bousfield localization in the presence of multiplicative structure. Our goal is to place an emphasis on examples of various types. This is an expository article, written to be part of an upcoming book.

The project of Greenlees et al. on understanding rational G-spectra in terms of algebraic categories has had many successes, classifying rational G-spectra for finite groups, SO(2), O(2), SO(3), free and cofree G-spectra as well as rational toral G-spectra for arbitrary compact Lie groups. This paper provides an introduction to the subject in two parts. The first discusses rational G-Mackey functors, the action of the Burnside ring and change of group functors. It gives a complete proof of the well-known classification of rational Mackey functors for finite G. The second part discusses the methods and tools from equivariant stable homotopy theory needed to obtain algebraic models for rational G-spectra. It gives a summary of the key steps in the classification of rational G-spectrain terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to clearly see the analogy between the algebraic and topological classifications.

These notes give a brief introduction to the category of spectra as defined in stable homotopy theory. In particular, Section 5 discusses an extensive list of examples of spectra whose properties have been found to be interesting.