55 problems

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…

Isomorphism Conjecture. For the chosen $K$- or $L$-theory, the assembly map $A_{X_G,G/H}$ should be a weak equivalence of spectra.

Let $\Gamma$ be a discrete group, let $\underline{E}\Gamma$ be its classifying space for proper actions, let $K_*^\Gamma(\underline{E}\Gamma)$ denote equivariant $K$-homology, and…

Let ${\mathcal H}^?_*$ be an equivariant homology theory with values in $R$-modules, let $G$ be a group, and let ${\mathcal C}$ be a family of subgroups of $G$. Let…

Rationalized Farrell-Jones conjecture. For every group $G$ and every $n\in\mathbb{Z}$, this rationalized Farrell-Jones assembly map is an isomorphism.

Bass's Whitehead-group conjecture. The group $\mathit{Wh}(G)$ vanishes.

Let $G$ be a group, let $R$ be a ring, and consider the assembly map … Here $BG$ is the classifying space of $G$, $K(R)$ is the algebraic $K$-theory spectrum of $R$, and $R[G]$ is…

Let $G$ be a group, let $d53c$ be a family of subgroups of $G$, let $E$ be a functor from small $d53d$-linear categories to spectra that sends category equivalences to weak equival…

Weinberger–Yu's conjecture. The group $\mathcal M_{\gamma_1,\ldots,\gamma_n}$ has rank $n$, and every nonzero element of it lies outside the image of the assembly map

Let $\mathcal F$ be one of the stable topological pseudo-isotopy, algebraic $K$-theory, or $L$-theory functors, with orientation data for the $L$-theory case. Let $\Gamma$ be a dis…

Farrell–Jones Nil-summand conjecture. In general, the cokernel of these assembly maps is conjectured to be a “sum” of $Nil$s. The statement records a pattern attributed to Farrell…

Let $\pi$ be a group, and let $A^L_\pi$ and $A^K_\pi$ denote the assembly maps in $L$-theory and $K$-theory, respectively. Their rationalizations are obtained by tensoring with the…

Fibered homotopy K-theory isomorphism conjecture. The group $G$ satisfies the (fibered) $\mathit{KH}$-isomorphism conjecture for $R$ if the pair $(G,\mathcal{FIN})$ satisfies the (…

Finite-dimensional strong Borel conjecture. If $B\pi$ is finite-dimensional, then this assembly map is an isomorphism. This is presented as a more modest version of the strong Bore…

Let $Gamma$ be a discrete group, let $BGamma$ be its classifying space, and let $mu$ be the assembly map. The rational assembly map is … The Strong Novikov conjecture. The map…

Let $cGamma$ be a group, let ${cE}$ be a $cGamma$-space universal for virtually cyclic subgroups, and write ${cB}={cE}/cGamma$. For a space $X$ with a free, properly discontinuous…

Fibered Isomorphism Conjecture. The map

Farrell–Jones and Baum–Connes assembly conjectures. The maps $\operatorname{asmb}_i$ for $i=1,2,3$ are bijective. These comprise the rational and real assembly maps associated with…

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…

Bounded-geometry positive scalar curvature exact-sequence conjecture. There is a commutative diagram

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…

Let $G$ be a group, let $R$ be a ring with involution, and let $\mathbf{L}^{\langle-\infty\rangle}(R)$ be the associated $L$-theory spectrum, whose $n$th homotopy group is…

Let $G$ be a group, let $R$ be a regular ring, and let $BG$ be a classifying space for $G$. The spectrum $\mathbf{K}(R)$ represents algebraic $K$-theory, and the assembly map is th…