14 problems

Let $G$ be a topological group with a universal proper space $\mathcal E G$. Let $A$ be a $C^*$-algebra and let $B$ be a $G$-algebra. Define the naive topological $KK$-groups by ……

Analytic equivariant coarse Novikov conjecture. The composition of $\pi_*$ and the assembly map, namely the Miščenko–Kasparov assembly map $\nu_X^{\Gamma}$, is injective. This is a…

Let $\Gamma$ be a countable discrete group acting properly on a proper metric space $X$. Let $KX_*^{\Gamma}(X)$ denote the universal $K^{\Gamma}$-group assembling the equivariant i…

Equivariant coarse Novikov conjecture. The map $\mu_X^{\Gamma}$ is rationally injective. This is the injectivity part of the equivariant coarse Baum–Connes conjecture and implies t…

Shifted Bernoulli full-shift probability conjecture.

The coarse geometric $\ell^p$-Novikov conjecture. If $\Gamma$ is a discrete metric space with bounded geometry, then the index map $e_*$ is injective.

Finite-part assembly conjecture. The group $\widetilde{\mathcal M}_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of $K_0^{\mathrm{fin}}(C^*(\Gamma))$ lies out…

Analytic Novikov conjecture. The map $e$ induces an injection

Strong Novikov conjecture. The map $e$ induces an injection

Let $\Sigma$ be a Cantor set with a minimal action of $\mathbb{Z}^p$ that preserves a Borel probability measure $\mu$. Let $\sigma$ be the multiplier on $\mathbb{Z}^p$ associated t…

Let $G$ be a group with torsion elements $g_1,g_2,\ldots,g_k$ of distinct positive orders $d_1,\ldots,d_k$. For each $g_i$, let $p_i$ be the idempotent corresponding to $g_i$, and…

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…

Let $G$ be a countable group. For an element $g\in G$ of finite order $d$, define … in $\mathbb{Q}G$, and let $K_0^{\mathrm{fin}}(C^*(G))$ be the subgroup of $K_0(C^*(G))$ generate…

The $\ell^1$ Bass conjecture. The image of $HS^1$ is contained in