11 problems

Gromov's rational homology vanishing conjecture. The induced map on fundamental classes satisfies

For each $k\geq 1$, let $\mu_k\in H_{4k}(\mathrm{Out}\,F_{2k+2};\mathbb{Q})$ be the homology classes introduced through the trace maps. Morita's conjecture. These classes are all t…

Let $C_{p,d}(\mathbb{P}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space, let $L_qH_k(-)_{\mathbb{Q}}$ denote Lawson homology…

Multi-node filling conjecture. $\mathbb{Q}$-homology disk symplectic fillings cannot exist for resolution graphs with two or more nodes.

Completeness conjecture. The known examples give a complete list of surface singularities with $\mathbb{Q}$-homology disk smoothings.

Let $M$ be a compact connected oriented smooth manifold of dimension $n$, let $93=c0_1(M,m_0)$ for a chosen basepoint $m_0$, and let $bd:M\rightarrow B93$ be the classifying map to…

Fix $m\geq 2$, $k\geq 1$, and $n\geq 3$. Let … be the graphing map. Graphing injectivity conjecture. The induced maps on rational homotopy and rational homology are injective: … ar…

Let $X$ be a closed manifold, and let $d4db(X)$ denote its free loop space. Its rational homology is $H_\ast(\mathcal{L}(X);\mathbb{Q})$. Gromov's conjecture. The rational homology…

Let $R$ denote the radius of convergence of the Poincaré series of the space of long knots modulo immersions. Positive-radius conjecture. The radius of convergence is positive; tha…

Let $\mathcal{K}^{m\bullet}_d$ be the cosimplicial space of long links of $m$ strands in $\mathbb{R}^d$ constructed by Munson and Volić, and let its homology Bousfield–Kan spectral…

For each $k\geq 1$, the construction gives a class … corresponding under Kontsevich's isomorphism to the displayed primitive cohomology class. Non-triviality conjecture. These clas…