59 problems

Let $\mathcal{X}$ be a collection of topological spaces, and let $\prod\mathcal{X}$ denote the collection of spaces homotopy equivalent to a finite-type product of spaces in…

Homotopy-commutativity conjecture. The diagram

Let $\Gamma$ be the operad whose $k$-th space is the nerve of the category of permutations of $\{1,2,\dots,k\}$, and let $\Gamma^{(n)}$ be Smith's filtration, in which a simplex be…

Let $\mathbf{X}$ be a space and write $G_{\ast}=\Omega^{-1}\pi_{\ast}\mathbf{X}$. A free simplicial $\Pi$-algebra resolution is a free simplicial $\Pi$-algebra resolution of…

For the space $\Omega^2_{-1}$ of locally convex curves in $\mathbb{S}^3$ with the indicated endpoint data, let $\Omega^2_{-1}$ denote the relevant space and consider its non-contra…

Strong Barratt exponent conjecture. The double loop map

Let $X$ and $W$ be simply-connected spaces with the homotopy type of finite-type CW-complexes, and let $R$ be a coefficient ring. Suppose that $H_*({\Omega X};R)$ is torsion-free.…

Let $\mathbb P^n$ be complex projective space, let $L^2\mathbb P^n$ denote its double loop space, and let $\hol{}(n)$ and $\rat{}(n)$ denote the holomorphic and rational mapping sp…

Let $\nu$ be the map constructed from the configuration-space model of the loop space of the classifying space $B\mathsf{G}$, using the maps between configurations, evaluation fibe…

Millson–Zombro conjecture. There exists a $PGL(2,\mathbb{C})$-equivariant embedding

Let $M$ be a closed simply connected manifold of dimension $n$, let $k_{p}$ denote the coefficient field associated with a prime $p$, and consider the partial sums … Here…

Let $S^n$ be the round $n$-sphere, let $\tau$ be its antipodal map, and define the twisted loop space … Let $\phi^d_\infty$ denote the wrapped Dehn twist on $T^*S^n$ obtained by co…

Let $\operatorname{Fl}_{n+1}$ be the flag variety, let $\beta$ be a strictly monotonic class, and write $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ for the corresponding algebrai…

Hodge decomposition conjecture.

Cotangent-fiber Hodge microsheaf conjecture. Hain's Hodge structure is induced by a Hodge microsheaf whose underlying microsheaf corresponds to a cotangent fiber.

Rivera–Takeda–Wang's conjecture. For an appropriate choice of the lift of $\bbDelta(1)$, the secondary coproduct coincides with the loop coproduct; in particular, the appropriate c…

Let $\mathcal{K}$ be a simplicial complex and let $\mathbf{k}$ be a ring such that $H_*(\Omega\mathrm{DJ}(\mathcal{K});\mathbf{k})$ is a free $\mathbf{k}$-module. Recall that a Hop…

Let $X$ and $Y$ be based cellular spaces, with $X$ compact, and let $Y^X$ denote the based continuous maps from $X$ to $Y$. For an integer $r\geq 0$, say that a map $a\in Y^X$ is…

Let $M$ be a closed orientable manifold of dimension $n$, let $\Lambda$ be its free loop space, and let $\Lambda_0\subset\Lambda$ be the subspace of constant loops. Write $\mu$ for…

Let $n>3$ and let $q otin Z(operatorname{Quat}{n+1})$. Consider the space ${cal L}n(1;q)$ of locally convex curves with endpoint data $(1,q)$. If it contains convex curves, let…

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…

Let $M$ be a manifold with Euler characteristic $\chi$, let $\Lambda$ be its free loop space, and let $\eta$ denote the coefficient system used to define loop homology. For a basep…

Charming-map conjecture. The following statements are true:

Let $X$ be a manifold with de Rham differential graded algebra $\Omega^*(X)$, and let $HC^\lambda_*(\Omega^*(X))$ denote the Connes version of cyclic cohomology of this de Rham com…

Let $G$ be a connected Lie group in the class considered by Rezk, let $X$ be a finite $G$-CW complex, and let $E_G^*(X)$ denote Rezk's equivariant cohomology theory constructed fro…