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We study a certain construction designed to bring together the following two topics: $i$) Dyer--Lashof-operations in negative Tate-cohomology, $ii$) the description of negative Tate-cohomology in terms of joins. It has the merit of making (some) sense in a more general context: where the loop-space of the space under study no longer has to be of compact homotopy-type. The exposition given here is a streamlined version of a previous version of mine, available here at the Arxiv.

Four Opponent Powers Practice of Purifying Negative Actions To purify our negative karma we must practice the four opponent powers: the power of regret, the power of reliance, the power of remedy, and the power of resolve. Within these four, one begins by reflecting on regret; the awareness that actions we have committed bring suffering to ourselves and others. We further identify the reliance (or refuge) of one such as the Buddha, who inspires us with his example; with the power of remedy we commit ourselves to a practice which alleviates the results of the actions we have done. We complete with the power of resolve resolving not to repeat those actions which bring suffering to ourselves and others. Lam rim, Stages of the Path, Wisdom of Emptiness, Bodhicitta, Bodhisattva, Regret, Karma, Causes and Conditions, Interdependent Origination, Refuge, Remedy, Resolve, Promise

Four Opponent Powers Practice of Purifying Negative Actions To purify our negative karma we must practice the four opponent powers: the power of regret, the power of reliance, the power of remedy, and the power of resolve. Within these four, one begins by reflecting on regret; the awareness that actions we have committed bring suffering to ourselves and others. We further identify the reliance (or refuge) of one such as the Buddha, who inspires us with his example; with the power of remedy we commit ourselves to a practice which alleviates the results of the actions we have done. We complete with the power of resolve resolving not to repeat those actions which bring suffering to ourselves and others. Lam rim, Stages of the Path, Wisdom of Emptiness, Bodhicitta, Bodhisattva, Regret, Karma, Causes and Conditions, Interdependent Origination, Refuge, Remedy, Resolve, Promise

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, almost maximal or maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient of a product of spheres of dimensions greater than or equal to three. As an immediate consequence, we prove the Maximal Symmetry Rank Conjecture for all $M\in \mathcal{M}_{0}^n$. Finally, we show the Maximal Symmetry Rank Conjecture holds for dimensions less than or equal to nine without assuming the torus action is almost maximal or maximal.

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, almost maximal or maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient of a product of spheres of dimensions greater than or equal to three. As an immediate consequence, we prove the Maximal Symmetry Rank Conjecture for all $M\in \mathcal{M}_{0}^n$. Finally, we show the Maximal Symmetry Rank Conjecture holds for dimensions less than or equal to nine without assuming the torus action is almost maximal or maximal.