Mathematics

We compute the homotopy groups of the C 2 fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a TMF -module, it is isomorphic to the tensor product of TMF with an explicit finite cell complex.

We show that the C 2 -equivariant and R -motivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author.

In previous work, we related homotopy types of (G,n) -complexes when G has periodic cohomology to projective ZG modules representing the Swan finiteness obstruction. We use this to determine when X∨ S n ≃Y∨ S n implies X≃Y for (G,n) -complexes X and Y , and give lower bounds for the number of minimal homotopy types of (G,n) -complexes when this fails. The proof involves constructing projective ZG modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case n=2 , difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem.

Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from these resolutions. We also give an alternative description of the morphism sets in terms of Yoneda Ext groups.

We define combinatorial counterparts to the geometric string vertices of Sen-Zwiebach and Costello-Zwiebach, which are certain closed subsets of the moduli spaces of curves. Our combinatorial vertices contain the same information as the geometric ones, are effectively computable, and act on the Hochschild chains of a cyclic A ∞ -algebra. This is the first in a series of two papers where we define enumerative invariants associated to a pair consisting of a cyclic A ∞ -algebra and a splitting of the Hodge filtration on its cyclic homology. These invariants conjecturally generalize the Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants from symplectic geometry, and the Bershadsky-Cecotti-Ooguri-Vafa invariants from holomorphic geometry.

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.

We construct an explicit equivariant cellular decomposition of the (4n−1) -sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an S 3 -equivariant decomposition of the flag manifold of S L 3 (R) .

A generalised Postnikov tower for a space X is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to X . In this paper we give a characterisation of a polyhedral product Z K (X,A) whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include p -local and rational versions of the theorem. We end with a group theoretic application.

In this article we study the construction of characteristic classes for principal G -bundles equipped with an additional structure called transitionally commutative structure (TC structure). These structures classify, up to homotopy, possible trivializations of a principal G -bundle, such that the induced cocycle have functions that commute in the intersections of their domains. We focus mainly on the cases where the structural group G equals SU(n) , U(n) or Sp(n) . Our approach is an algebraic-geometric construction that relies on the so called power maps defined on the space B com G , the classifying space for commutativity in the group G .

The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber M depend only on the underlying ? M -fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for ? M -fibrations, Baut( ? M ) , and its cohomology ring, i.e., the ring of characteristic classes of ? M -fibrations. For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable ξ -fibration together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(ξ) as well as the subring generated by the generalized Miller-Morita-Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given ? M -fibration comes from a manifold bundle.