A Counterexample to Cohomological Rigidity of Toric Manifolds

The main result determines the zeroth integral Milnor-Witt stem of the motivic sphere spectrum in the Morel-Voevodsky motivic stable homotopy category of the integers. The component in weight zero is the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms over the integers. Along the way, cellularity of connective Witt theory, as well as an absolute purity result for the η-inverted motivic sphere spectrum, is established over Dedekind domains of mixed characteristic.

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arXiv2026

Realization of permutation modules as homology of CW-complexes

In this paper, we investigate the realizability problem, which asks whether prescribed group actions on graded modules can be realized by the groups of self-homotopy equivalences of CW-complexes acting on their homology. We provide a partial answer in the case of arbitrary groups and permutation modules concentrated in certain degrees. As a consequence, we realize every group as the group of self-homotopy equivalences of an $R$-local CW-complex with arbitrary prescribed connectivity, where $R$ may be chosen with $ρ(R)$ sufficiently large. In particular, this provides a complete answer to Kahn's realizability problem.

math.AT↗

arXiv2026

Braidings of Self-Equivalences and Bordism

Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of ${\mathcal E}(M(\ell))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton--Kreck for closed, oriented $4$-manifolds.

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