11 problems

Let $M^n$ be a closed oriented manifold, with fundamental class $[M]\in H_n(M,\mathbb Z)$ and total $L$-class $\mathcal L(M)\in H^*(M,\mathbb Q)$. Let $\pi$ be a discrete group, le…

Novikov's conjecture. The higher signatures are oriented homotopy invariant. The conjecture is a central problem in index theory and topology. In the paper's setting, it is related…

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…

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…

Let $(M,\partial M)$ be a compact oriented smooth manifold with boundary, and let $G$ and $\Gamma$ be groups associated to the boundary and manifold through a homomorphism…

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

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

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

Let $Y\subseteq X$ be aspherical compact spaces, and let … be the maximal relative Baum–Connes map. If $j\colon \pi_1(Y)\to \pi_1(X)$ is injective, let … be the reduced relative Ba…

Let $B\pi$ be the classifying space of a discrete group $\pi$, and consider the assembly map … Here $C^*(\pi)$ is the group $C^*$-algebra. Strong Novikov conjecture. The map…

Strong Novikov conjecture. The analytic assembly map $\alpha$ is a monomorphism. This is the injectivity statement connecting $KO$-homology of the classifying space to the $K$-theo…