194 problems

Let $G$ be a torsion-free group, and let $\operatorname{Wh}(G)$ denote its Whitehead group. Whitehead conjecture. The Whitehead group of any torsion-free group should vanish: … Thi…

Let $d$ be a nonnegative integer, and let $X$ be a nonempty finite free $(\mathbb{Z}/2)^d$-CW complex. Its homology is taken with coefficients in $\mathbb{F}_2$. Carlsson's conject…

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

Let $X$ be a compact Kähler manifold with infinite fundamental group, and let $\widetilde{X}$ be its universal cover. Carlson–Toledo conjecture. There exists a finite étale cover…

Wedge-of-spheres conjecture. For all $k\geq 2$, the complex $\mathsf{Cliq}_{k}(G)$ is homotopy equivalent to a wedge sum of $(2k-3)$-dimensional spheres. Moreover, the number of sp…

Kahle–Newman conjecture. The distribution of the Sylow $p$-subgroup of $H_1(C_n)$ follows the Cohen–Lenstra distribution.

Friedlander–Milnor reformulation. The Friedlander–Milnor conjecture for $\operatorname{SL}_2$ over $\overline{\mathbb{Q}}$ can be reformulated as

Let $G$ be a Lie group with maximal torus $T$, and let $G\to G/T$ be the quotient fibration. Write $E_r^{*,*}(G;\mathbb{Z})$ for its Leray–Serre spectral sequence, and regard…

Triangulation-invariance conjecture. There is a good definition of $T_p(M,G)$, and

Let $P=\mathbb F_2[t_1,\ldots,t_h]$ be the polynomial algebra on $h$ generators, regarded as a module over the mod-two Steenrod algebra $\mathcal A$, and let … be its degree-$n$ co…

Łuczak–Peled conjecture. For every $d \geq 2$ and $p=p(n)$ such that $|np-c_d|$ is bounded away from $0$, $H_{d-1}(Y_d(n,p))$ is torsion-free asymptotically almost surely.

Let an oriented matroid be a combinatorial analogue of a real vector space, and let an OM-Grassmannian be the finite simplicial complex associated analogously to the Grassmannian o…

Let $G$ be the group considered in the paper, let $BG$ be its classifying space, and let $\mathcal{L}BG$ denote its free loop space. Write $G(\mathbb{F}_q)$ for the corresponding f…

Fix a finite prime field $\mathbb{F}_p$ and let $\mathcal{A}_p$ denote the Steenrod algebra. For a topological space $X$, write $H^*(X)$ for its cohomology with coefficients in…

Stiefel's conjecture. The class $w_{n-i}(M)$ can be defined simply as the sum of all $i$-simplices in the barycentric subdivision of a triangulation of $M$.

Let $\operatorname{\mathbb{R}P}^n$ denote real projective $n$-space, and let its length mean the invariant defined in the paper. Length monotonicity conjecture. The length of … is…

Let $(Y,y)$ be a finite aspherical CW-complex which is not acyclic, and let $f\colon (X,x)\to (Y,y)$ be a continuous map. Define the degree of $f$ by letting $\deg(f)$ be the least…

Let $M$ and $N$ be manifolds, let $f:M\rightarrow N$ be a continuous map, and let ${\text{Map}}(M,N;f)$ denote the space of maps homotopic to $f$. Fix a basepoint $*$ in $M$, and d…

Let a graph property be a partition of the unlabeled graphs on a fixed number of vertices into graphs with and without the property. The property is monotone if it is preserved und…

Let … denote the $k$th Betti number of a random neighborhood or clique complex, with edge probability $p$. Unimodality conjecture. For both random neighborhood and clique complexes…

Zaidenberg's isotriviality conjecture. If $X$ is contractible, then $p$ is an isotrivial polynomial: its generic fibres are pairwise isomorphic.

Let $K$ be a $1$-reduced simplicial set and let $EK$ denote its suspension. The cobar construction $\tom\widetilde C(EK)$ admits a natural, strict Alexander–Whitney coalgebra struc…

Operadic computability conjecture. Operadic techniques should, in finite time, solve the same subproblem addressed by the other proposed approaches: they should organise the…

Canonical product conjecture. This canonical product always induces the correct product on $H^*(X)$, making the above identification an isomorphism of algebras.

Finite subset-space cell-structure conjecture. The space $\exp_k X$ has a cell structure obtained from $\exp_{k-1} X$ by adding cells of dimensions $k-c\leq i\leq nk$.