{'fraction_non_alphanumeric': 0.05015197568389058, 'fraction_numerical': 0.00911854103343465, 'mean_word_length': 4.045977011494253, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On models of cohomology rings of spaces with a torus action

A model of a graded algebra A∗ is defined to be a graded differential algebra B∗, dB such that H[B∗, dB ] = A∗. In this article we construct models for rational cohomology rings of topological spaces with a torus action and Dolbeault cohomology rings of complex manifolds with a holomorphic torus action. These models turn out to be a useful tool for attacking Halperin’s conjecture on toral rank [1] and for studying complex geometry of non-Kähler manifolds (see, for instance, [2]). We correct mistakes made in the construction of similar models in [2] (Proposition 5.1) and [3] (Lemma 1.2) by proving the existence of corresponding models under the additional condition of the formality of the homotopy orbit space. Also, we give an example of a non-formal topological space whose cohomology ring is not described by this model. In the proof of Theorem 1 and the construction of the example (Theorem 2) we use techniques of rational homotopy theory (see [4], [5]). Let X be a topological space with a torus T m = (S) action, let XT m := X×T m ET m be its Borel construction, or homotopy orbit space, and let f : XT m → BT m be the classifying map of the principal T -bundle X × ET m → XT m . Note that if the action T m : X is free, then XT m ≃he X/T .