Matthias Franz: Papers


  1. (with Jeffrey D. Carlson) A product on the two-sided bar construction, appendix to: Jeffrey D. Carlson, The cohomology of biquotients via a product on the two-sided bar construction, arxiv:2106.02986
  2. (with Xin Fu) Cohomology of smooth toric varieties: naturality, arxiv:2104.03825
  3. Szczarba's twisting cochain is comultiplicative, arxiv:2008.08943
  4. Dga models for moment-angle complexes, arxiv:2006.01571, to appear in Fields Institute Communications

Published articles

  1. The cohomology rings of homogeneous spaces, J. Topol. 14 (2021), 1396-1447
  2. Homotopy Gerstenhaber formality of Davis-Januszkiewicz spaces, Homology Homotopy Appl. 23 (2021), 325-347
  3. The cohomology rings of smooth toric varieties and quotients of moment-angle complexes, Geom. Topol. 25 (2021), 2109–2144
  4. The cohomology rings of real toric spaces and smooth real toric varieties, Proc. Roy. Soc. Edinburgh Sect. A (2021), online
  5. (with Christopher Allday and Volker Puppe) Syzygies in equivariant cohomology in positive characteristic, Forum Math. 33 (2021), 547-567
  6. Szczarba's twisting cochain and the Eilenberg-Zilber maps, Collect. Math. 72 (2021), 569-586
  7. Homotopy Gerstenhaber algebras are strongly homotopy commutative, J. Homotopy Relat. Struct. 15 (2020), 557-595
  8. (with Jianing Huang) The syzygy order of big polygon spaces, Alg. Geom. Top. 20 (2020), 2657-2675
  9. (with Hitoshi Yamanaka) Graph equivariant cohomological rigidity for GKM graphs, Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), 107-110
  10. 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 II.4.3 in Smith's paper [19].
  11. (with Santiago López de Medrano and John Malik) Mutants of compactified representations revisited, Bol. Soc. Mat. Mex. 23 (2017), 511-526
  12. 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, including all main results, is not affected.
  13. 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
  14. 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.
  15. (with Christopher Allday and Volker Puppe) Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property, Alg. Geom. Top. 14 (2014), 1339-1375
    One should add to the assumptions of Corollary 3.8 that the coefficient field be of characteristic 0.
  16. (with Christopher Allday and Volker Puppe) Equivariant cohomology, syzygies and orbit structure, Trans. Amer. Math. Soc. 366 (2014), 6567-6589
  17. (with Anthony Bahri and Nigel Ray) Weighted projective spaces and iterated Thom spaces, Osaka J. Math. 51 (2014), 89-121
  18. (with Anthony Bahri, Dietrich Notbohm and Nigel Ray) The classification of weighted projective spaces, Fund. Math. 220 (2013), 217-226
  19. Tensor products of homotopy Gerstenhaber algebras, Homology Homotopy Appl. 13 (2011), 249-262
  20. (with Volker Puppe) Exact sequences for equivariantly formal spaces, C. R. Math. Acad. Sci. Soc. R. Can. 33 (2011), 1-10
  21. Describing toric varieties and their equivariant cohomology, Colloq. Math. 121 (2010), 1-16
    The XΣ in Theorem 1.1 denotes the toric variety defined by the fan Σ.
  22. (with Anthony Bahri and Nigel Ray) The equivariant cohomology ring of weighted projective space, Math. Proc. Cambridge Philos. Soc. 146 (2009), 395-405
  23. (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
    A simpler construction of the mutant Z2 appears in Section 7.1 of the paper "Big polygon spaces". This is generalized to all mutants in the paper "Mutants of compactified representations revisited".
  24. (with Volker Puppe) Exact cohomology sequences with integral coefficients for torus actions, Transformation Groups 12 (2007), 65-76
  25. (with Frédéric Bihan, Clint McCrory and Joost van Hamel) Is every toric variety an M-variety? Manuscripta Math. 120 (2006), 217-232
  26. (with Volker Puppe) Steenrod squares on conjugation spaces, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 187-190
  27. Comment on Novel public key encryption technique based on multiple chaotic systems, Phys. Rev. Lett. 96 (2006), 069401
  28. Koszul duality and equivariant cohomology, Documenta Math. 11 (2006), 243-259
  29. The integral cohomology of toric manifolds, Proc. Steklov Inst. Math. 252 (2006), 53-62
    A better title would have been "The integral cohomology of smooth toric varieties". Smooth (not necessarily compact) toric varieties are equivariantly homotopy-equivalent to partial quotients of moment-angle complexes.
  30. (with Andrzej Weber) Weights in cohomology and the Eilenberg-Moore spectral sequence, Ann. Inst. Fourier 55 (2005), 673-691
  31. Koszul duality and equivariant cohomology for tori, Int. Math. Res. Not. 42 (2003), 2255-2303
  32. 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 valid 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. A substantial generalization can be found in the paper "The cohomology rings of smooth toric varieties and quotients of moment-angle complexes".


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

Matthias Franz, 2022-03-17