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Transformation Groups

, Volume 22, Issue 4, pp 933–965 | Cite as

A QUOTIENT CRITERION FOR SYZYGIES IN EQUIVARIANT COHOMOLOGY

  • MATTHIAS FRANZEmail author
Article

Abstract

Let X be a manifold with an action of a torus T such that all isotropy groups are connected and satisfying some other mild hypotheses. We provide a necessary and sufficient criterion for the equivariant cohomology H T * (X) with real coefficients to be a certain syzygy as module over H*(BT). It turns out that, possibly after blowing up the non-free part of the action, this only depends on the orbit space X/T together with its stratification by orbit type. Our criterion unifies and generalizes results of many authors about the freeness and torsion-freeness of equivariant cohomology for various classes of T-manifolds.

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References

  1. 1.
    C. Allday, M. Franz, V. Puppe, Equivariant cohomology, syzygies and orbit structure, Trans. Amer. Math. Soc. 366 (2014), 6567–6589.Google Scholar
  2. 2.
    C. Allday, M. Franz, V. Puppe, Equivariant Poincare–Alexander–Lefschetz duality and the Cohen–Macaulay property, Alg. Geom. Top. 14 (2014), 1339–1375.Google Scholar
  3. 3.
    C. Allday, V. Puppe, Cohomological methods in transformation groups, Cambridge Univ. Press, Cambridge, 1993.Google Scholar
  4. 4.
    A. Ayzenberg, M. Masuda, S. Park, H. Zeng, Cohomology of toric origami manifolds with acyclic proper faces, J. Symplectic Geom., to appear, arXiv:1407.0764v3 (2015).Google Scholar
  5. 5.
    G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup, Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) 54 (2002), 1–41.Google Scholar
  6. 6.
    E. Bifet, C. De Concini, C. Procesi, Cohomology of regular embeddings, Adv. Math. 82 (1990), 1–34.Google Scholar
  7. 7.
    G. E. Bredon, The free part of a torus action and related numerical equalities, Duke Math. J. 41 (1974), 843–854.Google Scholar
  8. 8.
    M. Brown, Locally at imbeddings of topological manifolds, Ann. Math. (2) 75 (1962), 331–341.Google Scholar
  9. 9.
    W. Bruns, J. Herzog, Cohen–Macaulay Rings, 2nd ed., Cambridge Univ. Press, Cambridge 1998.Google Scholar
  10. 10.
    W. Bruns, U. Vetter, Determinantal Rings, Lecture Notes in Mathematics, Vol. 1327, Springer, Berlin, 1988.Google Scholar
  11. 11.
    T. Chang, T. Skjelbred, The topological Schur lemma and related results, Ann. Math. (2) 100 (1974), 307–321.Google Scholar
  12. 12.
    M. W. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417–451.Google Scholar
  13. 13.
    D. Eisenbud, The Geometry of Syzygies, Springer, New York, 2005.Google Scholar
  14. 14.
    M. Franz, Big polygon spaces, Int. Math. Res. Not. 2015 (2015), 13379–13405.Google Scholar
  15. 15.
    M. Franz, Syzygies in equivariant cohomology for non-abelian Lie groups, to appear in: F. Callegaro et al. (eds.), Configuration Spaces (Cortona, 2014), Springer, Cham, 2016; arXiv:1409.0681v3.Google Scholar
  16. 16.
    M. Franz, V. Puppe, Freeness of equivariant cohomology and mutants of compactified representations, in: M. Harada et al. (eds.), Toric Topology (Osaka, 2006), Contemp. Math. 460, AMS, Providence, RI, 2008, pp. 87–98.Google Scholar
  17. 17.
    O. Goertsches, S. Rollenske, Torsion in equivariant cohomology and Cohen–Macaulay G-actions, Transform. Groups 16 (2011), 1063–1080.Google Scholar
  18. 18.
    O. Goertsches, D. Töben, Torus actions whose equivariant cohomology is Cohen–Macaulay, J. Topology 3 (2010), 819–846.Google Scholar
  19. 19.
    M. Goresky, R. Kottwitz, R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83.Google Scholar
  20. 20.
    P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.Google Scholar
  21. 21.
    T. S. Holm, A. R. Pires, The topology of toric origami manifolds, Math. Res. Lett. 20 (2013), 885–906.Google Scholar
  22. 22.
    W. S. Massey, Homology and Cohomology Theory, Dekker, New York, 1978.Google Scholar
  23. 23.
    M. Masuda, Cohomology of open torus manifolds, Proc. Steklov Inst. Math. 252 (2006), 146–154.Google Scholar
  24. 24.
    M. Masuda, T. Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), 711–746.Google Scholar
  25. 25.
    J. R. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), 113–128.Google Scholar
  26. 26.
    M. Mustaţặ, Local cohomology at monomial ideals, J. Symbolic Comput. 29 (2000), 709–720.Google Scholar
  27. 27.
    G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.Google Scholar
  28. 28.
    K. Yanagawa, Alexander duality for Stanley–Reisner rings and squarefreen -graded modules, J. Algebra 225 (2000), 630–645.Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada

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