We classify cohomogeneity one actions on smooth, simply connected, closed manifolds with the rational cohomology of a sphere. In particular, we show that such a manifold is diffeomorphic to a sphere, a Brieskorn variety, the Wu manifold \operatorname{SU}(3)/{\operatorname{SO}}(3) , one of two homogeneous spaces of the form \mathbf{G}_{2}/{\operatorname{SU}}(2) , or a member of a particular four-parameter family of 7 -manifolds.
We show that a variant of Lurie's infinity-operads, where the category of finite pointed sets is replaced by that of spans of finite sets, can be identified as precisely the Lawvere theories of the analytic monads previously studied by Gepner, Kock, and the author. This gives an equivalence between infinity-operads (or more precisely a "flagged" or "pinned" version thereof) and analytic monads, with an infinity-operad O corresponding to the monad for O-algebras in spaces. In particular, the infinity-operad O is completely determined by this monad. To prove this we extend the equivalence between infinity-categorical Lawvere theories and algebraic monads, due to Gepner, Groth, and Nikolaus, to the many-object (or many-sorted) case.
Given a complexified Kähler class on a compact Kähler manifold, we introduce a generalisation of the K-energy functional, defined on the space of complexified Kähler potentials, whose critical points are solutions of the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. We prove that this extended K-energy is convex along geodesics in the space of almost calibrated potentials. As an application, we show that, at least in some notable cases, solutions of the scalar curvature equation with B-field are unique in their class, confirming the expectation that the scalar curvature equation with B-field identifies canonical representatives of complexified Kähler classes.
Using non-A(1)-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic K-theory of singular rings, such as rings of the form Z/p(n), Z[x]/(x(e)), and C(X; C) for X a compact Hausdorff space. These refinements are made possible by the use of integral p-adic Hodge theory, as a replacement for the standard use of trace methods in K-theory.
We study equivalences of the form \Sigma^{V}X\simeq\Sigma^{W}X , where G is a compact Lie group, X is a G -spectrum, and V and W are G -representations. These equivalences encode a periodicity phenomenon in G -equivariant homotopy theory which generalizes the classical James periodicity for G=C_{2} .In the case where X=C(a_{\lambda}) is the cofiber of an Euler class, we construct an \mathit{RO}(G) -graded J -homomorphism J\colon\pi_{\lambda}\mathit{KO}_{G}\rightarrow\pi_{\star}^{G} C(a_{\lambda})^{\times} which gives control over these periodicities. It also produces infinite periodic families in the G -equivariant stable stems. We illustrate this with several explicit examples.More generally, our work gives information about \mathit{RO}(G) -graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the G -homotopy fixed point spectral sequence, and other equivariant Atiyah–Hirzebruch spectral sequences.
