23 problems

Torus-knot conjecture. If $K$ is a nontrivial $SU(2)$-averse knot, then $K$ is a torus knot.

Let $M$ be an integer homology $3$-sphere, let $\widetilde{I}_n(M)$ be the decoration-independent unframed invariant obtained by subtracting the framing correction from $I_n(M)$, a…

Free-fundamental-group conjecture. Then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Quinn's conjecture. If $s(X^4)=0$, then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

Let $\pi$ be a group, and let $A^L_\pi$ and $A^K_\pi$ denote the assembly maps in $L$-theory and $K$-theory, respectively. Their rationalizations are obtained by tensoring with the…

H-ribbon characterization conjecture. A knot $K$ is h-ribbon with group $G$ if and only if there exists an epimorphism $\varphi\colon \pi_1(M_K)\twoheadrightarrow G$ such that cond…

An R-link is a link $L$ such that 0-surgery on $L$ yields a connected sum of copies of $S^1\times S^2$. A 0-framed unlink is an unlink whose components all have framing zero, and a…

Let $L=K_1\cup\dots\cup K_n$ be an $n$-component link in $S^3$. A 0-framed surgery on $L$ means surgery with framing zero on every component, and let $\#_n(S^2\times S^1)$ denote t…

Let $\pi$ be a finite $2$-group and let $N\triangleleft T\trianglelefteq W\triangleleft\pi$ be normal subgroups such that $\pi/N$ is cyclic, $T/N\cong\mathbb Z/{2}$, and…

For a closed oriented or non-orientable manifold product with an Arf invariant normal map, the resulting surgery obstruction modulo the ordinary Arf–Kervaire invariant is called a…

Let $M_{2p}$ be a closed topological nonorientable $4$-manifold with cyclic fundamental group of order $2p$, where $p>1$ is odd. Theorem Cancellation Homeomorphism Classes asserts…

Let $K$ be a knot, and let $S^3_r(K)$ denote the manifold obtained by surgery along $K$ with slope $r$. Purely Cosmetic Surgery Conjecture. If … via an orientation-preserving homeo…

Let $K\notin \mathcal{D}$ be a knot, and suppose that there is a knot $K'\in \mathcal{D}$ with … Here $\Delta_K(x)$ denotes the Alexander polynomial of $K$, and…

Let $K$notin $\mathcal{D}$ be a knot admitting positive alternating surgeries. Write $\mathcal{S}_{\rm alt}(K)$ for the set of alternating surgery slopes, and let $N$ denote the re…

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $S_x$ be the leaf through $x\in BG$. Let…

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $\mathcal{L}^{lf}$ be the cosheaf assigning to each…

Let $e$ be an embedding of a pochette into $S^4$, let $S_e$ denote its associated spun 2-knot, and let $S^4(e,p/q,\varepsilon)$ denote the result of pochette surgery with parameter…

Structure-set rank conjecture. The free ranks satisfy

Disc embedding conjecture. The circles $f_i(S^1)$ bound disjointly embedded discs in $M$ with transverse spheres, inducing the same framing on $f_i(S^1)$ as the $f_i$.

$s$-cobordism conjecture. There is a homeomorphism

Let $p\colon M\to B$ be a continuous map satisfying the standing assumptions: $M$ is a connected closed topological manifold; $B$ is an aspherical closed topological manifold admit…

$S^1$-stability conjecture. The manifold $M$ admits a metric of positive scalar curvature if and only if its product with the circle does:

Let $M$ be a compact oriented manifold, let $S(M)$ be its structure group, and let $S_0(M)$ be the subgroup generated by elements $[(f,M')]-[(\psi\circ f,M')]$, where $f:M'\to M$ i…