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Yigal Kamel

Publications and source records attributed to Yigal Kamel.

5 recordsLinked to original sources
arXiv2026

Maximal compatibility of disklike $G$-transfer systems

Transfer systems are a combinatorial model for $N_{\infty}$-operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they encode the additive transfers and multiplicative norms of a ring-type structure. In this paper, given a transfer system encoding an additive structure, we give explicit formulae for the maximal transfer system it is compatible with. Our formulae simplify for disklike transfer systems, which typically encode additive structures. Further, we prove that (maximal) compatibility is functorial with respect to the inflation map induced by a quotient of groups, letting us compute maximal compatible transfer systems as inflations of connected transfer systems.

math.AT↗
arXiv2025

The homotopy fixed points of Real spin bordism

We show that the 2-local splitting of spin$^c$ bordism by Anderson--Brown--Peterson and Stong refines to a $C_2$-equivariant map in the category of spectra with $C_2$-action from Real spin bordism to a sum of (higher) connective covers of $\mathrm{ku}_{\mathbb{R}}$ and suspensions of mod 2 Eilenberg--Mac Lane spectra. We use this to deduce a corresponding 2-local splitting of the homotopy fixed points of Real spin bordism. We also discuss prospects that arise in the genuine setting.

math.AT↗
arXiv2025

Real spin bordism and orientations of topological $\mathrm{K}$-theory

We construct a commutative orthogonal $C_2$-ring spectrum, $\mathrm{MSpin}^c_{\mathbb{R}}$, along with a $C_2$-$E_{\infty}$-orientation $\mathrm{MSpin}^c_{\mathbb{R}} \to \mathrm{KU}_{\mathbb{R}}$ of Atiyah's Real K-theory. Further, we define $E_{\infty}$-maps $\mathrm{MSpin} \to (\mathrm{MSpin}^c_{\mathbb{R}})^{C_2}$ and $\mathrm{MU}_{\mathbb{R}} \to \mathrm{MSpin}^c_{\mathbb{R}}$, which are used to recover the three well-known orientations of topological $\mathrm{K}$-theory, $\mathrm{MSpin}^c \to \mathrm{KU}$, $\mathrm{MSpin} \to \mathrm{KO}$, and $\mathrm{MU}_{\mathbb{R}} \to \mathrm{KU}_{\mathbb{R}}$, from the map $\mathrm{MSpin}^c_{\mathbb{R}} \to \mathrm{KU}_{\mathbb{R}}$. We also show that the integrality of the $\hat{A}$-genus on spin manifolds provides an obstruction for the fixed points $(\mathrm{MSpin}^c_{\mathbb{R}})^{C_2}$ to be equivalent to $\mathrm{MSpin}$, using the Mackey functor structure of $\underlineπ_*\mathrm{MSpin}^c_{\mathbb{R}}$. In particular, the usual map $\mathrm{MSpin} \to \mathrm{MSpin}^c$ does not arise as the inclusion of fixed points for any $C_2$-$E_{\infty}$-ring spectrum.

math.AT↗
arXiv2021

Generalizations and strengthenings of Ryser's conjecture

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $ν(H)$, the vertex cover number is at most $(r-1)ν(H)$. This far reaching generalization of König's theorem is only known to be true for $r\leq 3$, or $ν(G)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $α(G)$, there exists at most $(r-1)α(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

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