An introduction to filtered and synthetic spectra

In this paper, we develop a bordism-theoretic approach to rigidity problems for toric and quasitoric manifolds. We prove that two toric manifolds are isomorphic as varieties if and only if they are weakly equivariantly unitary bordant. We also establish a parallel rigidity result for omnioriented quasitoric manifolds satisfying the injectivity condition, showing that their equivariant unitary bordism classes completely determine their omniorientation-preserving equivariant homeomorphism types. Thus, equivariant bordism provides a topological framework for detecting geometric and combinatorial rigidity.

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arXiv2026

Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

math.AT↗

arXiv2026

Cyclic ABC Massey Products

This paper refines the notion of cyclic Massey products to the bi-graded setting, just as quadruple ABC Massey products refine the notion of quadruple Massey products. The result, we call ``cyclic ABC Massey products,'' are in general non-trivial and contain information different from the quadruple ABC Massey products.

math.AT↗