43 problems

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…

Goda–Teragaito lens space surgery conjecture. Then $K$ is a fibered knot and

Let a surgery on a knot in a three-manifold produce a lens space. In the context above, non-torus knots admit at most two such surgeries, and these slopes must be successive intege…

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 the 4-dimensional topological surgery conjecture be the assertion that, for a 4-dimensional Poincaré pair $(X,N)$, the sequence … is exact. In the link formulation, 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 $M$ 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…

Let $\pi$ be a finite group that acts freely on $S^{2n-1}$ for some $n$. An action on an odd-dimensional sphere is almost linear if its restriction to every cyclic or $2$-hyperelem…

Farrell–Jones Nil-summand conjecture. In general, the cokernel of these assembly maps is conjectured to be a “sum” of $Nil$s. The statement records a pattern attributed to Farrell…

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…

Let $(W^5,M_0,M_1)$ be a $5$-dimensional topological s-cobordism. s-cobordism conjecture. Then $W$ is homeomorphic to the product … This is the four-dimensional topological s-cobor…

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…

Fixity propagation conjecture. An analogue of the main theorem should hold in this case: $G$ should act freely, smoothly, and homologically trivially on a product of spheres

Let $N$ be a closed manifold undergoing a surgery deformation through manifolds $N_{\tau}$, where $N_{\tau}$ is diffeomorphic to $N$ for $\tau<0$ and to the surgered manifold…

Let $L(3,2;\frac{c}{d})$ denote the link family used in the source, and let $L(3n\pm1,3;\frac{2}{3})$ and $L(2n\pm1,2;\frac{2}{3})$ be the corresponding GST link-family members. Th…

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…

Let $\kappa^h_j$ denote the surgery-theoretic obstruction maps discussed in the paper. Oozing conjecture. The original oozing conjecture asserts that $\kappa^h_j=0$ for every…

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 $(X,Y)$ be a Poincaré pair with $X$ aspherical and $Y$ a closed manifold. Borel Existence Conjecture. The relative structure set $\mathcal S(X\operatorname{rel}Y)$ is nonempty;…

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…