66 problems

Let $F$ be an elliptic space with minimal model $\Lambda V$, and let $F\to X\to B$ be a fibration. The condition that $\dim V^{\mathrm{odd}}=\dim V^{\mathrm{even}}$ is imposed on t…

Mixed Hodge structure conjecture. The coordinate ring and Lie algebra of $G_M$ should have natural mixed Hodge structures such that the homomorphisms

Let $A$ be a $1$-connected Poincaré duality CDGA of formal dimension $n$ odd. For each $\xi\in (A\otimes A)^{2n-2}$, let $C(\xi)$ be the associated CDGA, and let … be the natural s…

Let $M_n$ be the real locus of the moduli space of stable genus-zero curves with $n$ marked points. A connected space is rational $K(\pi,1)$ if its rational completion is a…

Halperin's conjecture. The fibration is TNCZ, meaning that $j$ induces a surjection on rational cohomology.

Let $(\Lambda V,d)$ be an elliptic minimal algebra whose differential has homogeneous length. Write … where $e$ is given by the formula of the theorem on homogeneous-length differe…

Let $X$ be a simply connected finite CW complex, and let $X_B$ denote the constant $\infty$-stack whose values are the Poincaré $\infty$-groupoid of $X$. Formality criterion conjec…

Configuration-space integral realization conjecture. There exists a map

Let $X$ be a simply-connected finite $CW$-complex. For a prime $p$ and $r\geq 1$, say that $X$ is mod-$p^r$ hyperbolic if the number of $p^r$-torsion summands in the homotopy group…

Let $X$ be a finite, simply-connected $CW$-complex. Call $X$ rationally elliptic if it has finitely many rational homotopy groups, and say that $X$ has a finite homotopy exponent a…

Let $M$ be a compact, simply connected manifold with an isometric action by a compact group $G$. The orbit space $M/G$ is assumed to have dimension two and to be covered by the hyp…

Let $M$ be a compact, simply connected manifold with a polar action by a compact group $G$, meaning an isometric action admitting a section $S$ with $\dim S=\dim M/G$ that meets al…

Let $M_{n}^{J}$ be the degree-$n$ piece at filtration level $J$, let $\mathrm{wt}(\omega)$ denote the weight of $\omega$, and let $d_{\sim}$ be the indicated component of the differ…

Toral rank and Carlsson conjectures.

Moore's conjecture. The following are equivalent:

Poincaré-polynomial conjecture. There exist integers $k,s,n_i,m_j$ such that

Bott conjecture. The manifold $M$ is rationally elliptic.

Libgober's conjecture. If $X$ is rationally elliptic, then its mixed Hodge structure is of Hodge–Tate type; equivalently, every non-zero mixed Hodge number $h^{k,p,q}$ satisfies

Hilali conjecture. If $X$ is a rationally elliptic space, then

Let $F\to E\overset{p}\longrightarrow B$ be a fibration of simply connected spaces of finite type. Suppose that $\Omega p$ has a homotopy section and that $E$ is coformal over a ri…

Let $X$ be a simply connected space and let $\mathcal{L}X=\operatorname{Map}(S^1,X)$ be its free loop space. Write $\Omega X$ for the based loop space, and call $X$ rationally hype…

Equivariant model conjecture. The fixed subalgebra $A^{\Phi}$ is a model for $X$.

Let $X$ be a space such that $H^*(X;\mathbb{Q})$ is finite-dimensional, and let … The rational Gottlieb elements of $X$ are the elements associated to the rational Gottlieb group o…

Let $X$ be a formal space with bigraded Sullivan minimal model $(\land V,d)$, where … and the differential satisfies $d(V_0)=0$ and $d(V_i)\subseteq (\land V)_{i-1}$ for $i\geq 1$.…

Let $\mathbb{B}_k$, for $0 \leq k \leq 8$, and $\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ be the indicated series of del Pezzo surfaces. Let $\mathcal{L}^{k}_c S^4$, fo…