Torus actions on homotopy complex projective spaces

Abstract

We prove that an effective action of a torus T on a homotopy CP m is linear if m < 4 · rk(T ) − 1. Examples show that the bound is optimal. Combining this with a theorem of Hattori we conclude that the total Pontrjagin class of such a manifold is given by the usual formula (1 + x 2 ) m+1 .

Torus actions on manifolds of positive sectional curvature

“…The Petrie conjecture [20] asks whether the total Pontrjagin class of M 2m is standard, i.e., whether it is given by (1+x2) re+l, where xcH2(M, Z) is a generator. Dessai and Wilking [5] show that the conclusion of the Petrie conjecture holds if M 2"~ supports an effective smooth action by a d-dimensional torus with d> 1(m+1). More precisely, it is shown that such an action is linear in the sense of Petrie [19].…”

Positive curvature and symmetry in small dimensions

“…The following result provides a sharp calculation of the Euler characteristic and the second and third homology groups of a positively curved 14-manifold in the presence of T 4 symmetry. Note that, if T 4 is replaced by T 5 in this statement, then M is tangentially homotopy equivalent to S 14 or CP 7 (see [Wil03,DW04]). We proceed to the proof.…”

The Weighted Connection and Sectional Curvature for Manifolds With Density

“…In [KW17, Theorem D], the first two authors prove a weak version of Wilking's theorem that does not rely on Grove and Searle's equivariant classification. Equipped with Theorem 3.10, together with the connectedness lemma and other results of [KW17], in the weighted setting, we are able to fully recover Wilking's classification (see [Wil03,DW04]).…”