4 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…

Let $\theta=(B,p,xi)$ be a family of $Rd$-bundles, and let T^% denote the covariant part of the transformation \tau^% associated to the extended $A$-theory characteristic. Here…

Bordered-manifold Hopf brushing conjecture. Let $W$ be a compact bordered manifold (first $C^{\infty}$ and then $C^0$). Assume that $c7(W)=0$, and when $W$ is odd-dimensional that…

Topological Hopf brushing conjecture. A closed topological manifold with vanishing Euler characteristic admits a brushing.