Torus equivariant algebraic models and compact realization
[Modèles algébriques équivariants de tore et réalisation compacte]
Zoller, Leopold1
1 Mathematisches Institut Theresienstr. 39 80333 Munich (Germany)
Annales de l'Institut Fourier, Online first, 58 p.

Soit T un tore compact. Nous prouvons que, jusqu’à l’équivalence rationnelle équivariante, la catégorie des espaces T-simplement connectés, de type T-fini avec un nombre fini de types d’isotropie est complètement décrite par certains systèmes finis d’algèbres graduées différentielles commutatives avec des choix cohérents de classes de cohomologie de degré 2. Nous montrons que les systèmes algébriques correspondant aux complexes T-CW finis sont exactement ceux qui satisfont la condition nécessaire imposée par le théorème de localisation de Borel ainsi que certaines conditions de finitude. Nous dérivons une caractérisation algébrique pour savoir quand une algèbre sur un anneau polyonmial est réalisée comme la cohomologie rationnelle équivariante d’un T-CW-complexe fini. Comme applications supplémentaires, nous prouvons que toute cohomologie de graphe GKM est réalisée par un complexe T-CW fini et nous classifions les algèbres de cohomologie équivariante de complexes S 1 -CW finis avec des points fixes discrets.

Let T be a compact torus. We prove that, up to equivariant rational equivalence, the category of T-simply connected, T-finite type T-spaces with finitely many isotropy types is completely described by certain finite systems of commutative differential graded algebras with consistent choices of degree 2 cohomology classes. We show that the algebraic systems corresponding to finite T-CW-complexes are exactly those which satisfy the necessary condition imposed by the Borel localization theorem along with certain finiteness conditions. We derive an algebraic characterization of when an algebra over a polyonmial ring is realized as the rational equivariant cohomology of a finite T-CW-complex. As further applications we prove that any GKM graph cohomology is realized by a finite T-CW-complex and classify equivariant cohomology algebras of finite S 1 -CW-complexes with discrete fixed points.

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DOI : 10.5802/aif.3618
Classification : 55N91, 55P62, 57S10
Keywords: Equivariant rational homotopy theory, Torus actions, Equivariant cohomology, Finite $T$-CW complex, Realization.
Mot clés : Théorie de l’homotopie rationnelle équivariante, actions de tore, cohomologie équivariante, complexe $T$-CW fini, réalisation.
Zoller, Leopold 1

1 Mathematisches Institut Theresienstr. 39 80333 Munich (Germany) 
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Zoller, Leopold. Torus equivariant algebraic models and compact realization. Annales de l'Institut Fourier, Online first, 58 p.

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