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Wednesday, 4 November 2026: 2 new papers, 4 cross-lists and 8 replacements in Algebraic Topology (math.AT), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Fix an odd prime $p$ and let $A$ be the mod $p$ Steenrod algebra. There is a normal sub-Hopf algebra $D$ of $A$, and we show that the cohomology of $A$ is $F$-isomorphic to the subalgebra of the cohomology of $D$ which is invariant under the $A$-action. We also show that the sub-Hopf algebras of $A$ with bounded profile function detect nilpotence in cohomology.
For any odd prime number $p$, we solve the topological period-index problem for a $p$-primary element of the Brauer group of a CW-complex of dimension not greater than $2p+2$.
We study the Farrell--Jones conjecture for graph products of pairs and for fundamental groups of polyhedral products. We introduce a pair-deletion decomposition and a local notion of FJ-admissibility, and prove a vertexwise combination theorem for graph products of pairs. The criterion applies, in particular, to ordinary graph products, free-factor pairs, suitable normal pairs, and classes controlled by acylindrical Bass--Serre actions. For polyhedral products, we combine these results with their description by relative graph products and graph products of pairs. We also obtain bounds and exact computations for amenable category, together with applications to simplicial volume and Borel rigidity of closed aspherical polyhedral-product manifolds.
We construct an endofunctor on (small) simplicial sets which freely adjoins a zero object to a (small) infinity-category, as the co-cone construction freely adjoins a terminal object. We then prove that this endofunctor induces a left adjoint to the inclusion of the infinity-category of (small) pointed infinity-categories and morphisms that preserve zero objects into the infinity-category of (small) infinity-categories. This adjunction, in turn, leads to the definition of a coherent choice of nullhomotopies on a pointed infinity-category. A coherent choice of nullhomotopies is a non-canonical, coherent choice of "higher zeroes" on a pointed infinity-category, mirroring the obvious definition of higher zeroes on the differential graded nerve of a differential graded category of chain complexes over an additive category. We then show that coherent choices of nullhomotopies give an alternate characterization of fiber sequences in pointed infinity-categories, generalizing the 1-categorical definition of kernels in terms of morphisms that compose to zero.
We compare linking-form bounds with the smooth and locally flat nonorientable 4-genera of low-crossing knots. For every prime knot with at most 13 crossings whose Witt-level linking-form bound is at least 2, this bound equals both genera. We also exhibit 41 prime knots with 12 or 13 crossings that bound locally flat M\"obius bands but no smooth ones, and whose smooth nonorientable 4-genus is 2. Our methods combine linking forms, Heegaard Floer correction terms and definite-lattice obstructions with explicit band constructions. They determine 3,300 of the 3,525 values left unknown in KnotInfo on 2 November 2026; the remaining 225 knots have range $[1,2]$. We obtain new lower bounds of 2 for 2,102 knots with 14 crossings and correct previously reported values for three knots with 11 crossings.
We study the tensor triangular geometry of derived categories of valuation domains, obtaining complete ring-theoretic characterizations of their main structural properties. In particular, although valuation domains are almost always non-noetherian, their derived categories always satisfy the local-to-global principle and hence the detection property. We establish precise criteria for homological detection, stratification, and homological stratification, revealing a clear hierarchy among these properties. Notably, we obtain examples of derived categories of commutative rings that are homologically stratified and satisfy the local-to-global principle, but are not stratified. We also determine when the category of modules over a valuation domain has Gabriel dimension, and use this to provide a negative answer to a question of Balchin, G\'omez and Stevenson. Overall, valuation domains provide a rich class of non-noetherian rings whose derived categories can be understood in considerable depth.