22 problems

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…

Rational strong Novikov conjecture. The group $\Gamma$ satisfies the rational strong Novikov conjecture if $\mu$ is injective. This is a rational injectivity condition on the assem…

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 $M$ be a compact oriented smooth manifold with boundary. Its relative higher signatures are the corresponding signature invariants for manifolds with boundary. Relative Novikov…

Coarse Baum–Connes conjecture. The rational Roe index map is an equivariant split monomorphism. This version is described as a further coarsening of the condition $FW$ and as imply…

Weinberger's conjecture. The homomorphism $\delta$ is an epimorphism. This is one of the coarse statements that imply a version of the Novikov conjecture; the supplied text gives n…

Let $M$ be as above, let $M^\prime\to M$ be the induced connected normal $\Gamma^\prime$-covering, and let $G^\prime$ be the group of pairs $(\phi,g)$ satisfying…

Analytic Novikov conjecture. The analytic assembly map $A_{\Gamma}$ is injective. The paper places this assertion in the descent from coarse Baum–Connes methods to the Novikov conj…

Novikov conjecture. The assembly map $A$ is injective for torsion-free groups. This is presented as a form of the Novikov conjecture and, via the preceding corollary, would imply t…

Gromov's conjecture. Rationally essential manifolds do not admit PSC metrics.

Let $G$ be a group, let $G_{\mathbf{Fin}}\mathbf{Orb}$ denote the orbit category restricted to finite subgroups, and consider the functor … The complex version of the Novikov conje…

Maximal rational strong Novikov conjecture. The group $\Gamma$ satisfies the maximal rational strong Novikov conjecture if $\mu_{\max}$ is injective. The paper notes that the assem…

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…

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

Strong Novikov conjecture. Under these hypotheses, the Dirac operator on $X$ twisted by some flat unitary Hilbert bundle over $X$ has nonzero kernel.

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

Let $G$ be a second countable locally compact Hausdorff group, let $X$ be a $G$-compact proper complete $G$-Riemannian manifold, and let $X'$ be another complete Riemannian $G$-com…

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…

Loday's algebraic $K$-theory Novikov conjecture. This assembly map is rationally injective. The conjecture concerns homotopy-theoretic detection of algebraic $K$-theory classes of…

Let $\Gamma$ be a group, let $R$ be an H-unital ring, and let $B\Gamma$ be the classifying space of $\Gamma$. Let $H_n(B\Gamma,\mathbb K(R)^{-\infty})$ denote the corresponding hom…

Consider the authors' program for proving the BC Novikov conjecture for foliations, in which the coefficient bundle has a leafwise flat structure and the second step establishes le…

Let $M$ and $M'$ be closed oriented manifolds with oriented foliations $F$ and $F'$, let $\varphi:(M',F')\to(M,F)$ be an oriented, leafwise oriented, leafwise homotopy equivalence,…