10 problems

Spanier–Whitehead duality conjecture. Reversing the ordering of the critical sets of an arbitrary framed flow category, while preserving the stable normal framing, results in takin…

Let $(\Theta,\Phi)$ be a tangential pair, let $F$ be the fibre of $\Theta \to \Phi$, let $A$ be the Thom ring spectrum of the index bundle over $\Omega F$, and let $R$ be the Thom…

Let $\Theta$ be a tangential structure, let $R$ be its associated ring spectrum, and suppose $\operatorname{Flow}^{\Theta} \simeq \operatorname{Perf}({}_R\operatorname{mod}_R)$. Fo…

Let $\Theta$ be a tangential structure and let $R$ be the ring spectrum defined from it. Let $\operatorname{Flow}^{\Theta}$ denote the category of $\Theta$-oriented flow categories…

Let $F$ be a space and let $R:= \operatorname{Thom}(\Omega F \to BO \times \mathbb{Z})$ be the associated unital ring spectrum. Let $\operatorname{Flow}^F$ denote the stable…

Let $\mathcal{F}$ and $\mathcal{G}$ be flow categories, and let $\mathcal{W}:\mathcal{F}\to\mathcal{G}$ be a $\tau_{\leq n}$-morphism. The paper proves the obstruction criterion wh…

Fix an integer $n$. Let $\Flow_{\tau_{\leq n}}$ be the category of flow categories with morphisms given by homotopy classes of $\tau_{\leq n}$-morphisms, where $\tau_{\leq n}$ deno…

Let $\Flow^{fr}$ be the homotopy category of framed flow categories, whose objects are framed flow categories and whose morphisms are framed morphisms up to framed bordism. Abouzai…

Flow category moves conjecture. If two framed flow categories determine the same stable homotopy type, then they are move equivalent.

Let $(\mathscr{C}_1,\iota_1,\varphi_1)$ and $(\mathscr{C}_2,\iota_2,\varphi_2)$ be framed flow categories. Their associated stable homotopy types are the stable homotopy types dete…