32 problems

Let $N$ be a rational homology sphere that is the link of an isolated complex singularity, and let $q_{min}$ denote the intersection form of its minimal resolution. For any negativ…

Intersection-lattice conjecture. The conclusion of Corollary still holds, under the same assumptions, if $X$ is allowed to range over the set of all symplectic fillings of $Y$.

Extension conjecture. The intersection form on $\pi_2M$ is extended from the integers.

Non-spin-simple class conjecture. If $N$ has an odd intersection form

Let $(V,0)$ be a hypersurface surface singularity, and let its Milnor fibre carry its intersection form with signature defined in the usual way. Durfee's conjecture. The signature…

Definiteness conjecture. This intersection form should be definite. In particular, the signature of $M$ should be non-positive if $n$ is odd and non-negative if $n$ is even.

The $10/8$-conjecture. Then

Let $X$ be a closed smooth spin topological four-manifold. For an appropriately chosen basis of its second homology, suppose its intersection matrix has the form … where $E_8$ is t…

Let $X$ be a simply connected closed smooth spin $4$-manifold, and write its even unimodular intersection form as … where $E_8$ is the positive-definite even form of rank $8$ and…

ALE gluing conjecture. The resulting closed $4$-manifold should have intersection form equivalent to that of

Let $f(z_0,\ldots,z_n)=z_0^{a_0}+z_0z_1^{a_1}+\cdots+z_{n-1}z_n^{a_n}$ with $n\geq1$. Define $r_k=a_0a_1\cdots a_k$, $r_{-1}=1$, and let $c_0,\ldots,c_\mu$ be the coefficients in…

Let $M$ be the Riemannian $4$-manifold and let the cycles in Theorem 1 be the cycles whose intersection pairing is being estimated. Incompressibility conjecture. The condition that…

Intersection-pairing conjecture. There exists a function $D(\cdot,\cdot,\cdot)$ such that, for any classes $[Q],[P]\in H_2(M,\mathbb Z)$,

11/8 conjecture. Every simply connected, closed, smooth $4$-manifold with even intersection form satisfies this inequality. This conjecture constrains which even unimodular forms c…

11/8 conjecture. A smooth structure on $X$ requires

Let $(P,\lambda)$ be a hermitian, sesquilinear form on a free $\mathbb{Z}[\mathbb{Z}]$-module of rank $2g$, where $g\in\{1,2\}$. A form is surface hyperbolic if it is isometric to…

Integral embedding criterion for joinpowers. An embedding

Let $Q$ be an integer intersection form realised by a smooth $4$-manifold with boundary $S^3$ and no $3$-handles. The 11/8 conjecture. The forms realised under this hypothesis are…

Let $Q$ be an integer intersection form that is realised by a smooth $4$-manifold with boundary $S^3$. The 11/8 conjecture. Every such form $Q$ is realised by a smooth $4$-dimensio…

For a $k$-complex $X$, define its ${\mathbb Z}$-rank $r_{\mathbb Z}(X)$ to be the minimum of $\operatorname{rk}\cap_{N,{\mathbb Z}}$ over all $(k-1)$-connected PL $2k$-manifolds…

Let $K$ be a $k$-complex and let $M$ be a PL $2k$-manifold. A map $h:K\to M$ is a ${\mathbb Z}_2$-embedding if the mod-$2$ intersection number of the images of every pair of non-ad…

Let $M$ be a closed oriented smooth $4$-manifold with indefinite even intersection form. Write $b_2(M)$ for its second Betti number and $\operatorname{sign}(M)$ for its signature.…

Let $X$ be a simply connected, boundaryless smooth $4$-manifold whose indefinite even intersection form is … Here $h$ and $a$ are integers, and $\mathbf{E}_8$ and $\mathbf{H}$ deno…

Let $p$ be a nonnegative integer and $q$ a natural number. The form … can be realized as the intersection form of a closed smooth spin $4$-manifold. Matsumoto's 11/8-conjecture. Th…

Trisected 11/8-conjecture. One has