19 problems

Let $(M,\omega)$ be a closed symplectic manifold, and let $(M,\omega)\rightarrow P\rightarrow B$ be a Hamiltonian fiber bundle, meaning that its structural group can be reduced to…

Let $Y$ be a complex analytic manifold and let $f:Y\to\Delta$ be a projective morphism, where $\Delta$ is a disc. A morphism is a homotopy fiber bundle if it has the homotopy-local…

Let $M$ be a compact Kähler manifold of dimension $k$, and let $E$ be a holomorphic $\mathbb{B}^n$-bundle over $M$ embedded in the associated $\mathbb{CP}^n$-bundle $\widehat{E}$.…

Associated-connection conjecture. The connection $\eta$ can be seen as an associated connection to a principal connection on $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$…

Let $F$ be a smooth manifold homeomorphic to … Such a manifold satisfies Property (S) when it has the spinnability property for fiber bundles defined earlier in the paper. Property…

Richmond–Slofstra's conjecture. The element $w$ has a Grassmannian BP decomposition.

Let $N\to B$ be a fiber bundle with base $B$ a connected closed manifold, and with all fibers diffeomorphic to a fixed connected closed manifold $Z$. Suppose there is no fiberwise…

Let $(V,\partial V)$ be a manifold with $\dim(V)\geq 3$. Let $X\hookrightarrow B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is a manifold with a po…

Fiber-bundle conjecture. If all points in $S$ are weakly $m$-strained, then $\phi:X\to S$ is a fiber bundle map.

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation i…

Torus-fibration conjecture. $M$ is finitely covered by a manifold diffeomorphic to a fiber bundle over an $m$-dimensional torus.

Let $(V,\partial V)$ be a manifold of dimension at least $3$. Let $X\to B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is any manifold with a point-s…

K-theory presentation conjecture. The ring $K^*(E(X))$ is a free $K^*(B)$-module of rank $m=\chi(X)$ and is isomorphic to $\mathfrak{S}/\mathfrak{J}$.

Let $M\to E\to B$ be a smooth fiber bundle whose fiber admits a negatively curved Riemannian metric. Assume that $B$ is a paracompact, Hausdorff space with the homotopy type of a s…

Let $X$ be a simply connected closed manifold or a simply connected finite simplicial complex, and let $p\colon E\to X$ be a fiber bundle with closed-manifold fiber $M$. Assume tha…

Let $X$ be a closed manifold or finite simplicial complex, and let $p\colon E\to X$ be a fiber homotopically trivial bundle with closed-manifold fiber $M$. Assume that $E$ admits a…

Let $X$ be a compact simply connected manifold or a simply connected finite simplicial complex, and let $M\to E\stackrel{p}{\to}X$ be a negatively curved bundle. A bundle is topolo…

Let $f:X\to Y$ be a proper morphism of complex spaces. It is a homotopy fiber bundle if locally over $Y$ each fiber inclusion is a homotopy equivalence; it is a ${\mathbb Z}$-homol…

Let $M\to B$ and $M'\to B$ be tangentially homeomorphic smooth manifold bundles with common fiber a closed oriented even-dimensional manifold, and let $\Theta(M',M)$ denote their s…