21 problems

Let $E^{p,q}_r(n)$ denote the unstable spectral sequence in the notation of the paper, indexed by the parameter $n$, and let ${}^V\!E^{p,q}_r$ denote the Vassiliev spectral sequenc…

Let ${\mathcal{SC}}_n$ be the Swiss cheese operad, let ${\mathcal E}_n$ be its second-color little-discs operad, and let…

Connected-sum homomorphism conjecture. The map $\pi_0(C_n)$ is a homomorphism from the monoid of knots, under connected sum, to the abelian group $\pi_0(AM_n({\mathbb I}^3))$.

2-fold loop-space conjecture. We conjecture that $AM_n({\mathbb I}^3)$ is always a $2$-fold loop space. Consequently,

Universal finite-type invariant conjecture. The map

Nontriviality conjecture. The element represented by $G(p,q,r)$ is non-trivial in

Budney-Conant-Koytcheff-Sinha conjecture. The map $\alpha_n$ is bijective for every $n$. This conjecture asserts that the finite-type information captured by $\mathsf{K}_n$ agrees…

Let $K_3$ be the space of long knots in $\mathbb{R}^3$. Let $U_{\mathcal{T}}$ and $\Sigma$ be the spaces in the preceding construction, and let … be the maps defined there. Let…

Let $O$ be a termwise $\Sigma$-finite, termwise cofibrant operad and let $R$ be a termwise $\Sigma$-finite, termwise cofibrant right module in $(\mathrm{Sp},\wedge)$. Write $K$ for…

Higher obstruction equivalence conjecture. Such data determines a point in

Let $I=[0,1]$, let $\Knots(I^3)=\Emb_\partial(I,I^3)$ be the space of long knots, and write $\KK(I^3):=\pi_0\Knots(I^3)$. For each $n\geq 1$, let $\pT_n(I^3)$ be the $n$th stage of…

Let $\mathcal{A}^I_{n-1}$ be the group of integral chord diagrams of degree $n-1$ modulo the relevant relations, let $AM_n$ be the $n$th stage of the Goodwillie–Weiss tower, and le…

Let $ev_n$ be the map from the knot space to the $n$th stage of the Goodwillie–Weiss tower, and let $E^2$ denote the second page of its spectral sequence. A singular knot determine…

Let $ev_n$ denote the map from the space of classical long knots to the $n$th stage of the Goodwillie–Weiss embedding-calculus tower. The tower has an $E^2$-page in total degree ze…

Let $m$ and $n$ be the dimensions indexing the little cubes operads, let $E_m^{\mathbb{1}}$ and $E_n^{\mathbb{1}}$ denote their unital versions, and let…

Multiple disjunction conjecture for Poincaré embeddings. If $r\geq2$, then the $r$-cube $E(Q_\bullet,N)$ is $(3-n+\Sigma)$-cartesian. This is one of the paper's conjectural extensi…

Let $N$ and $P$ be Poincaré spaces, and let $Q_1,\dots,Q_r$ be Poincaré spaces embedded in $N$. Let $\mathit{EF}(P,N-Q_\bullet)$ denote the $(r+1)$-cube obtained from the cube of P…

Poincaré embedding multiple disjunction conjecture. If the handle codimension restrictions are omitted and $r\geq1$, then the $r$-cube $E(P,N-Q_\bullet)$ is $(1-p+\Sigma)$-cartesia…

Let $P$ and $N$ be manifolds of dimensions $p$ and $n$, respectively, and let $f\colon P\to N$ be an immersion equipped with a lift $f_{j-1}\in E_{j-1}(P,N)$. For $j\geq 2$, let…

Let $N$ be a smooth compact $n$-dimensional manifold and assume $s\geq 2$. For $1\leq i\leq s$, let $(Q_i,\partial_0Q_i,\partial_1Q_i)$ be a smooth compact manifold triad, with the…

Let $N$ be a smooth compact $n$-dimensional manifold. Let $(Q_1,\dots,Q_r)$, with $r\geq 1$, be pairwise disjoint compact submanifolds of $N$ transverse to $\partial N$, and let…