121 problems

A fibration has trivial fibre transport when its fibre transport action is trivial, and a fibre is positively rationally elliptic when its rational homotopy and cohomology groups a…

A rationally elliptic space is a simply connected topological space $X$ such that … A smooth manifold is said to have nonnegative sectional curvature when it admits a Riemannian me…

Let $X$ be a finite-dimensional topological space, and let $T^k=(S^1)^k$ be a $k$-dimensional torus acting almost freely on $X$. The total Betti number is…

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…

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

Let $G$ be a compact Lie group. Write ${{\mathsf{Sp}}}^G_\mathbb{Q}$ for the category of rational $G$-spectra, and let $\EuScript{A}(G)$ be an algebraic category. Greenlees's algeb…

A compact, simply connected manifold $M$ is called rationally elliptic when … It is called rationally hyperbolic otherwise. Bott–Grove–Halperin conjecture. Every compact simply con…

Algebraic model conjecture. For each compact Lie group $G$, there is an abelian category $\mathcal{A}(G)$ and a Quillen equivalence

Lambrechts–Stanley model conjecture. The Lambrechts–Stanley model is a model for the configuration spaces of every simply-connected closed oriented manifold.

Gromov's conjecture. The homotopical distortion of $\alpha$ is $\Theta(L^n)$ if and only if the Hurewicz image of $\alpha$ is nonzero. Otherwise, the homotopical distortion is…

Let $X$ be a simply connected, finite CW complex. It is rationally hyperbolic if the sum of the loop-space Betti numbers … grows at least exponentially in $n$, and let…

A simply connected compact symplectic manifold is a compact symplectic manifold with trivial fundamental group. Lupton–Oprea conjecture. Every simply connected compact symplectic m…

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

Let $X$ and $Y$ be compact piecewise Riemannian spaces, and define … Assume that $Y$ is simply connected. Gromov's growth conjecture. The function $g_{[X,Y]}$ is always asymptotic…

Let $X$ and $Y$ be finite complexes, with $Y$ simply connected. For $L>0$, consider the mapping classes in $[X,Y]$ having representatives with Lipschitz constant at most $L$. Gromo…

Homotopy-generator conjecture.

Rational collapse conjecture. The spectral sequences for the pointed spaces rationally collapse in total degrees $2* \leq n-3$, while the corresponding spectral sequences for the u…

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…

Let $G/T$ be a flag manifold, let $F(n)$ be the complete flag manifold, and let $Map(G/T,F(n))$ denote the space of continuous maps $G/T\to F(n)$. For $f\in Map(G/T,F(n))$, let…

Ellipticity conjecture. If $M$ admits a Riemannian metric with zero topological entropy, then its universal covering has the rational homotopy type of a finite elliptic CW complex…

Let $L$ be the simplicial complex obtained from the simplex on $V$ by deleting all faces containing one or more of the pairwise disjoint subsets $\lambda(k)$, and let…

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…

Félix–Halperin–Thomas ellipticity conjecture. Manifolds of nonnegative sectional curvature are elliptic.

A simply connected closed Riemannian manifold is integrally elliptic if its integral homotopy is constrained in the sense intended by the integral ellipticity conjecture. Integral…

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…