10 problems

Universal finite-type invariant conjecture. The map

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…

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…

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…