11 problems

Let … be a 5-dimensional Seifert bundle with $L$ smooth. Assume that $L$ is a simply connected rational homology sphere. Rationality conjecture. Then $S$ is a rational surface. Thi…

Let $N$ be a rational homology sphere, and let the augmentation-free part of its Seiberg–Witten invariant be denoted by the corresponding augmentation-free invariant. Let the refin…

Let $N$ be a rational homology sphere. Seiberg–Witten–Frøy conjecture. For every rational homology sphere $N$ we have [the asserted formula is not present in the supplied source sp…

Let $Y$ be a rational homology sphere with $|H_1(Y;\mathbb Z)|=h$. Suppose that $Y$ bounds a negative-definite four-manifold $X$ whose first homology has no torsion. Write…

Let $L_f$ be a rational homology $7$-sphere arising as the link of a hypersurface singularity, and suppose the defining weighted homogeneous polynomial has degree…

The $g$-conjecture. The $g$-vector of a rational homology sphere is an $M$-vector.

Almost linear k-invariant conjecture. The group $G$ acts freely and homologically trivially on a rational homology $3$-sphere with almost linear $k$-invariant if and only if $G$ ac…

Let $Y$ be a rational homology 3-sphere. Let $\mathcal{I}(Y)$ denote the set of negative definite lattices that can be smoothly bounded by $Y$, modulo stable-equivalence, where two…

The sixth example consists of families of rational homology spheres of dimension $4n-1$ with $H_{2n}\approx \mathbb{Z}_3$, including links $K_{6l+2}$ and $K_{6l+4}$ related to…

For $m\geq 1$, let $A_2$ be the weight matrix defining the standard cross-polytope $\mathbb T^2$-action on $\mathbb C^4$, and let $Y_{mA_2}$ denote the quotient associated with the…

Let $A$ be an admissible $2\times n$ weight matrix homotopy equivalent to a matrix corresponding to a centered $(2k+1)$-gon with multiplicities, where $k\geq 1$, and let $X_A$ be t…