7 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…

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…

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $\mathcal{L}^{lf}$ be the cosheaf assigning to each…

Let $M$ be a closed oriented manifold with an effective $S^1$-action. For each connected component $F$ of $M^{S^1}$, assume the hypotheses on $\Gamma_F$, its smooth subalgebra, and…

Let $M$ be a smooth manifold and let $E\subset TM$ be an integrable subbundle defining a foliation $\mathcal{F}$ whose leaves are rationally contractible, meaning that for every le…

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 $M$ be a connected, closed, oriented, smooth manifold, let $L_M$ be its Hirzebruch $L$-class, and let $f:M\to B(\pi_1M)$ classify the universal cover. The class $f_*(L_M)$ belo…