25 problems

Let $M$ be a closed oriented manifold with an effective $S^1$-action. For each connected component $F$ of $M^{S^1}$, assume the hypotheses on $\Gamma_F$, its smooth subalgebra, and…

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a fixed-point-free differentiable circle action. For an orbit $O$, define its exceptional mu…

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a pseudo-free differentiable circle action. Let $O_1,\dots,O_k$ be the nonfree orbits with s…

Geometric Gamma-operation generation conjecture. $\Omega^{S^1}$ is generated over geometric versions of the operations $\Gamma_V$ by linear actions on projective spaces.

Projective-space generation conjecture. Stably complex $S^1$ actions with isolated fixed points up to bordism are generated by linear actions on projective spaces

Symplectic circle-action conjecture. Every symplectic 4-manifold which admits a circle action also admits (possibly a different) symplectic form and circle action which are symplec…

Kosniowski's conjecture. If $M^{2n}$ is not a boundary, then the number of fixed points is greater than $f(n)$. Kosniowski conjectured that the candidate function is

Let $M$ be a simply-connected, compact, smooth $4$-manifold with $H_2(M;\mathbb{Z})=\mathbb{Z}$, and suppose that its boundary is a connected sum … of lens spaces, where…

Dimension-bound conjecture. The dimension of $M$ is bounded above by a function of $k$; more precisely,

A pseudo-free differentiable circle action on the $5$-dimensional sphere is an action whose non-free orbits are isolated orbit types. Fintushel–Stern's Montgomery–Yang problem. Eve…

Let $T^1$ act isometrically and effectively on $X^4$, where $X^4$ is a $4$-dimensional, closed, positively curved, orientable Alexandrov space. Classification conjecture. Up to equ…

Let $M$ be a closed $4$-manifold. A connection on an associated sphere bundle is called positive definite when its induced symplectic structure has fibre normal degree $+2$. Fine's…

Let $G$ be a word-hyperbolic group whose ideal boundary is homeomorphic to a $2$-sphere, and suppose that $G$ acts faithfully on its boundary. Virtual COL2 conjecture. $G$ is virtu…

Let $G$ be a finitely generated torsion-free discrete subgroup of $\operatorname{Homeo_+}(S^1)$. A pants-like $\operatorname{COL}_2$ group has a pants-like collection of two invari…

Let $(X^6,\omega)$ be a closed symplectic six-manifold equipped with a symplectic $S^1$ action, and suppose that its fixed-point set contains at least one isolated fixed point. Ham…

Generalised Taubes conjecture. The singular locus $L$ is a meridionally fibered link.

Let $M$ be a closed $4$-manifold that admits a symplectic form and a smooth circle action. Baldridge's conjecture. $M$ also admits a symplectic circle action, possibly with respect…

Montgomery–Yang problem. Every pseudofree $\mathbb{S}^1$-action on $\mathbb{S}^5$ has at most $3$ non-free orbits.

Let $M$ be a simply connected positively curved 4-manifold equipped with an isometric circle action. A linear circle action is a circle action induced by a linear action on either…

Let $X$ be a closed symplectic $4$-manifold admitting a free $S^1$-action, and let $M$ be the orbit space of this action. Baldridge's fibering conjecture. Then the quotient manifol…

Let $\Gamma$ be any lattice in $\mathop{\mathrm{SO}}\nolimits(1,n)$. Virtual circle-action conjecture. Some finite-index subgroup of $\Gamma$ has a $C^\infty$ action on $S^1$ with…

Let $\Gamma$ be a lattice in $\mathop{\mathrm{SL}}\nolimits(n,\mathbb{R})$, with $n\geq 3$. Lattice action conjecture. $\Gamma$ has no faithful action on $S^1$. This asks whether W…

Let $M$ be an orientable closed simply connected nonnegatively curved $4$-manifold with an effective isometric $S^1$-action. Hsiang–Kleiner's conjecture. $M$ is diffeomorphic to ei…

Let $M$ be an orientable closed positively curved $4$-manifold with an effective isometric $S^1$-action. Hsiang–Kleiner's conjecture. $M$ is diffeomorphic to $S^4$ or ${\bf CP}^2$.…

Let $S$ be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least $2$, and let $G$ be a finite-index subgroup of the map…