35 problems

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

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

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…

Let $T^*$ be the braided Ptolemy-Thompson group, and let ${\rm Diff}^2_+(S^1)$ denote the group of orientation-preserving diffeomorphisms of the circle $S^1$ of class…

Isolated-fixed-point criterion. An $\mathbb S^1$-action is Hamiltonian if its fixed-point set is nonempty and isolated.

Let $E$ be a closed oriented 4--manifold admitting a symplectic structure and equipped with a free action of the circle $S^1$. The quotient $E \slash S^1$ is a 3--manifold. Baldrid…

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…

Vanishing conjecture.

Equivariant Kähler structure conjecture. Complexity one spaces admit an equivariant Kähler structure if and only if their painting is trivial.

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…

Let $M$ satisfy the assumptions of Theorem or Theorem; namely, in the first case, $M$ is a connected, oriented, closed manifold of even dimension with a semi-free $S^1$-action whos…

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 $X$ be a positively curved Riemannian orbifold of dimension $4$ with an isometric circle action. If the action has only isolated fixed points, then its underlying topological s…

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 $\Gamma_g$ be the fundamental group of a closed orientable surface of genus $g$, and let $\rho\colon \Gamma_g\to\operatorname{Homeo}_+(S^1)$ be a representation. Call $\rho$ ri…

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…

A tautly foliated $3$-manifold is a $3$-manifold equipped with a taut foliation, and a group is strictly $\operatorname{COL}_2$ when it is $\operatorname{COL}_2$ but not…

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…