Matthias Franz: Papers


I have created a page where I post up-to-date versions of the first four preprints below.
  1. The cohomology rings of smooth toric varieties and quotients of moment-angle complexes, arxiv:1907.047791
  2. Homotopy Gerstenhaber formality of Davis-Januszkiewicz spaces, arxiv:1907.04782
  3. Homotopy Gerstenhaber algebras are strongly homotopy commutative, arxiv:1907.04778
  4. The cohomology rings of homogeneous spaces, arxiv:1907.04777
  5. (with Jianing Huang) The syzygy order of big polygon spaces, arxiv:1904.01051, to appear in Alg. Geom. Top.
  6. (with Hitoshi Yamanaka) Graph equivariant cohomological rigidity for GKM graphs, arxiv:1710.08264v4, to appear in Proc. Japan Acad. Ser. A Math. Sci.

Published articles

  1. Symmetric products of equivariantly formal spaces, Canad. Math. Bull. 61 (2018), 272-281
    Proposition 2.2 of this article is actually a special case of Proposition 4.3 in Smith's paper [19].
  2. (with Santiago López de Medrano and John Malik) Mutants of compactified representations revisited, Bol. Soc. Mat. Mex. 23 (2017), 511-526
  3. A quotient criterion for syzygies in equivariant cohomology, Transformation Groups 22 (2017), 933-965. Correction, ibid. 24 (2019), 949-950
    Part (ii) of Proposition 3.3 from the original article has been corrected. The rest of the paper is not affected.
  4. Syzygies in equivariant cohomology for non-abelian Lie groups, pp. 325-360 in: Filippo Callegaro et al. (eds.), Configuration spaces (Cortona, 2014), Springer INdAM Ser. 14, Springer, Cham 2016
  5. Big polygon spaces, Int. Math. Res. Not. 2015 (2015), 13379-13405
    A Macaulay2 file with functions to compute the syzygy order of the equivariant cohomology of a big polygon space can be found here, and lists of all chambers in dimension at most 9 here. Puppe's argument mentioned after Lemma 4.4 is Lemma 3.12 in: Volker Puppe, Equivariant cohomology of (Z2)r‑manifolds and syzygies, Fund. Math. 243 (2018), 55-74.
  6. (with Christopher Allday and Volker Puppe) Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property, Alg. Geom. Top. 14 (2014), 1339-1375
  7. (with Christopher Allday and Volker Puppe) Equivariant cohomology, syzygies and orbit structure, Trans. Amer. Math. Soc. 366 (2014), 6567-6589
  8. (with Anthony Bahri and Nigel Ray) Weighted projective spaces and iterated Thom spaces, Osaka J. Math. 51 (2014), 89-121
  9. (with Anthony Bahri, Dietrich Notbohm and Nigel Ray) The classification of weighted projective spaces, Fund. Math. 220 (2013), 217-226
  10. Tensor products of homotopy Gerstenhaber algebras, Homology Homotopy Appl. 13 (2011), 249-262
  11. (with Volker Puppe) Exact sequences for equivariantly formal spaces, C. R. Math. Acad. Sci. Soc. R. Can. 33 (2011), 1-10
  12. Describing toric varieties and their equivariant cohomology, Colloq. Math. 121 (2010), 1-16
  13. (with Anthony Bahri and Nigel Ray) The equivariant cohomology ring of weighted projective space, Math. Proc. Cambridge Philos. Soc. 146 (2009), 395-405
  14. (with Volker Puppe) Freeness of equivariant cohomology and mutants of compactified representations, pp. 87-98 in: Megumi Harada et al. (eds.), Toric Topology (Osaka, 2006), Contemp. Math. 460, AMS, Providence 2008
    For a simpler construction of the mutant Z2 see Section 7.1 of the paper "Big polygon spaces". This is generalized to all mutants in the paper "Mutants of compactified representations revisited".
  15. (with Volker Puppe) Exact cohomology sequences with integral coefficients for torus actions, Transformation Groups 12 (2007), 65-76
  16. (with Frédéric Bihan, Clint McCrory and Joost van Hamel) Is every toric variety an M-variety? Manuscripta Math. 120 (2006), 217-232
  17. (with Volker Puppe) Steenrod squares on conjugation spaces, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 187-190
  18. Comment on Novel public key encryption technique based on multiple chaotic systems, Phys. Rev. Lett. 96 (2006), 069401
  19. Koszul duality and equivariant cohomology, Documenta Math. 11 (2006), 243-259
  20. The integral cohomology of toric manifolds, Proc. Steklov Inst. Math. 252 (2006), 53-62
  21. (with Andrzej Weber) Weights in cohomology and the Eilenberg-Moore spectral sequence, Ann. Inst. Fourier 55 (2005), 673-691
  22. Koszul duality and equivariant cohomology for tori, Int. Math. Res. Not. 42 (2003), 2255-2303
  23. Moment polytopes of projective G‑varieties and tensor products of symmetric group representations, J. Lie Theory 12 (2002), 539-549
    The polytope P(3,3,3) from the article in polymake format.

Unpublished notes

  1. The mod 2 cohomology ring of real moment-angle complexes (2018), 8 pages, pdf
  2. On the integral cohomology of smooth toric varieties, arXiv:math/0308253
    These notes contain the first correct proof for the cup product formula in the integral cohomology of moment-angle complexes. The result is phrased in terms of toric subvarieties of affine space, which are equivariantly homotopy-equivalent to moment-angle complexes.


  1. Koszul duality for tori, pdf
    doctoral dissertation supervised by Volker Puppe, Universität Konstanz 2001
  2. Darstellungstheorie und Quantenmechanik
    diploma thesis supervised by Volker Strassen, Universität Konstanz 1997, also Konstanzer Schriften in Mathematik und Informatik 186 (2003)