  In the 1930s, H. Hopf conjectured that a closed, even-dimensional manifold of
positive sectional curvature has positive Euler characteristic. We show this
under the additional assumption of an isometric $T^4$-action on the manifold,
improving from previous theorems of Kennard, Wiemeler and Wilking assuming a
$T^5$-action. More specifically, this is achieved by giving a rational
cohomology classification of possible fixed point components. The main new tool
is an improvement on the four-periodicity theorem originally developed by
Kennard through the use of characteristic class theory. As a second application
we give a rational cohomology classification of closed positively curved
even-dimensional manifolds without odd rational cohomology that admit an
isometric $T^6$-action.
