1,293 problems

Let $M$ be a topological manifold of dimension $3$ (a second-countable Hausdorff space each of whose points has a neighborhood homeomorphic to an open subset of $\mathbb{R}^3$). Su…

Does every compact complex structure on the smooth manifold $T^2\times S^4$ split biholomorphically as a product of complex manifolds? Equivalently, for every integrable complex st…

Is it true that for every co-sober topological space $X,$ the Hoare power space $\PH(X)$ is co-sober? In particular, does this hold when $X$ is $T_2$ or metrizab…

For every finite flag complex $\Delta$ on $n$ vertices and every pair of coefficient fields $K$ and $L$, is the seventh total Betti number of its Stanley--Reisner ring independent…

Let $M$ be a complete surface and let $f\colon M\to M$ be a $C^{1}$ diffeomorphism satisfying Condition A: $f$ has dense periodic points, there is a $Df$-invariant splitting…

Let $C$ be a nonempty closed, bounded, convex subset of a Banach space, let $f:C\to C$ be continuous, and suppose that $f^n$ is compact for some integer $n\ge 1$. Then $f$ has a fi…

For every commutative complex Banach algebra $A$, let $\mathfrak{M}(A)$ denote its Gelfand spectrum. Taylor's question asks whether, and how, each higher integral cohomology group…

For each integer $d\ge 1$, let $\mu_d$ be the minimum number of vertices among all simplicial complexes whose geometric realization is homeomorphic to real projective $d$-space…

For every torsion-free discrete group $G$, the first $^2$-Betti number satisfies $\beta^{(2)}_1(G)\leq \operatorname{nrk}(G)-1$, where…

Given a complete weighted Riemannian manifold $(M,g,e^{-f})$ with bounded potential $f$ and nonnegative Bakry–Émery Ricci curvature, namely…

For every integer $n\ge 1$ and every finite, nonzero Borel measure $\mu$ on $\mathbb{R}^n$ that is absolutely continuous with respect to Lebesgue measure, there exist $n$ affine hy…

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For every finite contractible $2$-dimensional polyhedron $P$, the product $P\times I$, where $I=[0,1]$, is collapsible to a point.

Let $X$ be a compact Hausdorff space. Call a linearly ordered set $(L,<)$ with its open-interval topology a linearly ordered topological space, and say that $X$ is a continuous ima…

A $3$-manifold is closed if it is compact and has empty boundary. A closed $3$-manifold $M$ is prime if whenever $M$ is homeomorphic to a connected sum $M_1 \# M_2$, one of…

Fix an integer $n\geq 1$. Let $\mathrm{Bord}_n^{\mathrm{fr}}$ denote the symmetric monoidal $(\infty,n)$-category of framed bordisms: its objects are $0$-dimensional framed manifol…

In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers whi…

In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is…

Ganea's conjecture is a now disproved claim in algebraic topology. It states that

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.

In differential geometry, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian…

Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic homology spheres of arbitraril…

Let $K \subset S^3$ be a knot, let $\gamma$ be a slope on $\partial N(K)$, where $N(K)$ is a tubular neighbourhood of $K$, and suppose that the Dehn surgery $S^3_\gamma(K)$ obtaine…

In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.

Let $\mathbf f:\mathbb R/2\pi\longrightarrow\mathbb C$ be a smooth unit-speed simple closed curve, and let $\mathcal D$ denote the space of deformations satisfying conditions (1)--…