T1: K-groups and cohomology

K-groups and cohomology groups are important invariants in different areas of mathematics, from arithmetic geometry to algebraic and geometric topology to operator algebras. The idea is to associate algebraic invariants to geometric objects, for example to schemes or stacks, C*-algebras, stable $\infty$-categories or topological spaces. Originating as tools to differentiate topological spaces, these groups have since been generalised to address complex questions in different areas. Cohomology theories such as étale, crystalline and prismatic are crucial in the structural study of schemes and the Langlands programme, while K-theory provides insights into algebraic structures and intersection theory but is also used in algebraic and geometric topology through applications in the study of manifolds and the classification of nuclear C*-algebras.

  • Mathematical fields

  • Collaborations with other topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Benjamin Brück, Jeremy Miller, Peter Patzt, Robin J. Sroka, and Jennifer C.H. Wilson. On the codimension-two cohomology of $SL_n(Z)$. Advances in Mathematics, 451:109795, August 2024. doi:10.1016/j.aim.2024.109795.

    $\bullet $ Christopher Deninger. Dynamical systems for arithmetic schemes. Indagationes Mathematicae, June 2024. doi:10.1016/j.indag.2024.05.007.

    $\bullet $ Thomas Nikolaus and Maria Yakerson. An alternative to spherical Witt vectors. arXiv e-prints, May 2024. arXiv:2405.09606.

    $\bullet $ Benjamin Antieau, Achim Krause, and Thomas Nikolaus. On the $K$-theory of $\mathbf {Z}/p^n$. arXiv e-prints, May 2024. arXiv:2405.04329.

    $\bullet $ Markus Land, Thomas Nikolaus, and Marco Schlichting. L-theory of $C^*$-algebras. Proceedings of the London Mathematical Society, 127(5):1451–1506, October 2023. doi:10.1112/plms.12564.

    $\bullet $ Benjamin Antieau, Achim Krause, and Thomas Nikolaus. Prismatic cohomology relative to δ-rings. arXiv e-prints, October 2023. arXiv:2310.12770.

    $\bullet $ Johannes Ebert. Diffeomorphisms of odd-dimensional discs, glued into a manifold. Algebraic & Geometric Topology, 23(5):2329–2345, July 2023. doi:10.2140/agt.2023.23.2329.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Recipes to compute the algebraic $\mathrm K$-theory of Hecke algebras of reductive $p$-adic groups. arXiv e-prints, June 2023. arXiv:2306.01510.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Inheritance properties of the Farrell-Jones conjecture for totally disconnected groups. arXiv e-prints, June 2023. arXiv:2306.01518.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Algebraic $\mathrm K$-theory of reductive $p$-adic groups. arXiv e-prints, June 2023. arXiv:2306.03452.

    $\bullet $ Eugen Hellmann. On the derived category of the Iwahori–Hecke algebra. Compositio Mathematica, 159(5):1042–1110, May 2023. doi:10.1112/S0010437X23007145.

    $\bullet $ Esmail Arasteh Rad and Urs Hartl. Category of $C$-motives over finite fields. J. Number Theory, 232:283–316, July 2022. doi:10.1016/j.jnt.2020.06.015.

    $\bullet $ Johannes Ebert and Jens Reinhold. Some rational homology computations for diffeomorphisms of odd-dimensional manifolds. arXiv e-prints, March 2022. arXiv:2203.03414.

    $\bullet $ Benjamin Antieau and Thomas Nikolaus. Cartier modules and cyclotomic spectra. J. Amer. Math. Soc., 34(1):1–78, January 2021. doi:10.1090/jams/951.

    $\bullet $ Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, and Wolfgang Steimle. Hermitian K-theory for stable ∞-categories III: Grothendieck-Witt groups of rings. arXiv e-prints, September 2020. arXiv:2009.07225.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Michael Borinsky, Benjamin Brück, and Thomas Willwacher. Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group. Selecta Mathematica, February 2025. doi:10.1007/s00029-025-01018-9.

    $\bullet $ Jörg Schürmann, Connor Simpson, and Botong Wang. A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells. Compositio Mathematica, 161(1):1–12, January 2025. doi:10.1112/s0010437x24007462.

    $\bullet $ Benjamin Brück, Yuri Santos Rego, and Robin J. Sroka. On the top-dimensional cohomology of arithmetic Chevalley groups. Proceedings of the American Mathematical Society, August 2024. doi:10.1090/proc/16948.

    $\bullet $ Benjamin Brück, Kevin I Piterman, and Volkmar Welker. The Common Basis Complex and the Partial Decomposition Poset. International Mathematics Research Notices, August 2024. doi:10.1093/imrn/rnae177.

    $\bullet $ Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, and Robin J. Sroka. Scissors automorphism groups and their homology. arXiv e-prints, August 2024. arXiv:2408.08081.

    $\bullet $ Linus Kramer and Raquel Murat García. Fibrations and coset spaces for locally compact groups. arXiv e-prints, August 2024. arXiv:2408.03843.

    $\bullet $ Benjamin Brück, Jeremy Miller, Peter Patzt, Robin J. Sroka, and Jennifer C.H. Wilson. On the codimension-two cohomology of $SL_n(Z)$. Advances in Mathematics, 451:109795, August 2024. doi:10.1016/j.aim.2024.109795.

    $\bullet $ Sylvain E. Cappell, Laurenţiu G. Maxim, Jörg Schürmann, and Julius L. Shaneson. Equivariant toric geometry and Euler-Maclaurin formulae—an overview. Rev. Roumaine Math. Pures Appl., 69(2):105–128, June 2024. doi:10.59277/RRMPA.2024.105.128.

    $\bullet $ Benjamin Brück and Robin J. Sroka. Apartment Classes of Integral Symplectic Groups. Journal of Topology and Analysis, June 2024. doi:10.1142/s1793525324500286.

    $\bullet $ Christopher Deninger. Dynamical systems for arithmetic schemes. Indagationes Mathematicae, June 2024. doi:10.1016/j.indag.2024.05.007.

    $\bullet $ Thomas Nikolaus and Maria Yakerson. An alternative to spherical Witt vectors. arXiv e-prints, May 2024. arXiv:2405.09606.

    $\bullet $ Calista Bernard, Jeremy Miller, and Robin J. Sroka. Partial bases and homological stability of $\rm GL_n$$(R)$ revisited. arXiv e-prints, May 2024. arXiv:2405.09998.

    $\bullet $ Shachar Carmeli, Thomas Nikolaus, and Allen Yuan. Maps between spherical group rings. arXiv e-prints, May 2024. arXiv:2405.06448.

    $\bullet $ Laurenţiu Maxim and Jörg Schürmann. Weighted Ehrhart theory via equivariant toric geometry. arXiv e-prints, May 2024. arXiv:2405.02900.

    $\bullet $ Benjamin Antieau, Achim Krause, and Thomas Nikolaus. On the $K$-theory of $\mathbf {Z}/p^n$. arXiv e-prints, May 2024. arXiv:2405.04329.

    $\bullet $ Benjamin Brück. (non-)Vanishing of high-dimensional group cohomology. arXiv e-prints, April 2024. arXiv:2404.15026.

    $\bullet $ Joachim Cuntz and James Gabe. Generalized homomorphisms and KK with extra structure. arXiv e-prints, April 2024. arXiv:2404.06840.

    $\bullet $ Ian Gleason and João Lourenço. Tubular neighborhoods of local models. Duke Mathematical Journal, March 2024. doi:10.1215/00127094-2023-0028.

    $\bullet $ Laurenţiu Maxim and Jörg Schürmann. Weighted Ehrhart theory via mixed Hodge modules on toric varieties. arXiv e-prints, March 2024. arXiv:2403.17747.

    $\bullet $ Johannes Ebert. Tautological classes and higher signatures. arXiv e-prints, March 2024. arXiv:2403.02755.

    $\bullet $ Paolo Aluffi, Leonardo Mihalcea, Jörg Schürmann, and Changjian Su. Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem. Annales Scientifiques de l'ÉNS, 57(1):87–141, February 2024. doi:10.24033/asens.2571.

    $\bullet $ Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, and Changjian Su. From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes. In A glimpse into geometric representation theory, volume 804 of Contemp. Math., pages 1–52. American Mathematical Society, 2024. doi:10.1090/conm/804/16110.

    $\bullet $ Johannes Ebert and Jens Reinhold. Some rational homology computations for diffeomorphisms of odd‐dimensional manifolds. Journal of Topology, January 2024. doi:10.1112/topo.12324.

    $\bullet $ Benjamin Brück, Francesco Fournier-Facio, and Clara Löh. Median quasimorphisms on $\rm CAT $$(0)$ cube complexes and their cup products. Geometriae Dedicata, 218(28):1–33, January 2024. doi:10.1007/s10711-023-00870-3.

    $\bullet $ Yifei Zhao. Half-integral levels. arXiv e-prints, December 2023. arXiv:2312.11058.

    $\bullet $ Markus Land, Thomas Nikolaus, and Marco Schlichting. L-theory of $C^*$-algebras. Proceedings of the London Mathematical Society, 127(5):1451–1506, October 2023. doi:10.1112/plms.12564.

    $\bullet $ Benjamin Antieau, Achim Krause, and Thomas Nikolaus. Prismatic cohomology relative to δ-rings. arXiv e-prints, October 2023. arXiv:2310.12770.

    $\bullet $ Markus Banagl, Jörg Schürmann, and Dominik J. Wrazidlo. Topological gysin coherence for algebraic characteristic classes of singular spaces. arXiv e-prints, October 2023. arXiv:2310.15042.

    $\bullet $ Urs Hartl and Yujie Xu. Uniformizing the moduli stacks of global $G$-shtukas II. arXiv e-prints, September 2023. arXiv:2309.17441.

    $\bullet $ Benjamin Brück and Zachary Himes. Top-degree rational cohomology in the symplectic group of a number ring. arXiv e-prints, September 2023. arXiv:2309.05456.

    $\bullet $ Thorben Kastenholz and Robin J. Sroka. Simplicial bounded cohomology and stability. arXiv e-prints, September 2023. arXiv:2309.05024.

    $\bullet $ Johannes Ebert. Diffeomorphisms of odd-dimensional discs, glued into a manifold. Algebraic & Geometric Topology, 23(5):2329–2345, July 2023. doi:10.2140/agt.2023.23.2329.

    $\bullet $ Lucas Mann and Annette Werner. Local systems on diamonds and $p$-adic vector bundles. International Mathematics Research Notices, 2023(15):12785–12850, July 2023. doi:10.1093/imrn/rnac182.

    $\bullet $ Benjamin Brück, Peter Patzt, and Robin J. Sroka. A presentation of symplectic Steinberg modules and cohomology of $Sp_2$$_n$$(\mathbb Z )$. arXiv e-prints, June 2023. arXiv:2306.03180.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Almost equivariant maps for td-groups. arXiv e-prints, June 2023. arXiv:2306.00727.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Recipes to compute the algebraic $\mathrm K$-theory of Hecke algebras of reductive $p$-adic groups. arXiv e-prints, June 2023. arXiv:2306.01510.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Inheritance properties of the Farrell-Jones conjecture for totally disconnected groups. arXiv e-prints, June 2023. arXiv:2306.01518.

    $\bullet $ Arthur Bartels and Wolfgang Lueck. Algebraic $\mathrm K$-theory of reductive $p$-adic groups. arXiv e-prints, June 2023. arXiv:2306.03452.

    $\bullet $ Christopher Deninger and Michael Wibmer. On the proalgebraic fundamental group of topological spaces and amalgamated products of affine group schemes. arXiv e-prints, June 2023. arXiv:2306.03296.

    $\bullet $ Lucas Mann. Normal and irreducible adic spaces, the openness of finite morphisms, and a Stein factorization. Nagoya Mathematical Journal, 250:498–510, June 2023. doi:10.1017/nmj.2022.40.

    $\bullet $ Eugen Hellmann. On the derived category of the Iwahori–Hecke algebra. Compositio Mathematica, 159(5):1042–1110, May 2023. doi:10.1112/S0010437X23007145.

    $\bullet $ Ben Heuer, Lucas Mann, and Annette Werner. The $p$-adic Corlette–Simpson correspondence for abeloids. Mathematische Annalen, 385(3):1639–1676, April 2023. doi:10.1007/s00208-022-02371-2.

    $\bullet $ Klaus Altmann, Amelie Flatt, and Lutz Hille. Extensions of toric line bundles. Mathematische Zeitschrift, 304(1):3, March 2023. doi:10.1007/s00209-023-03206-9.

    $\bullet $ Sylvain E. Cappell, Laurenţiu Maxim, Jörg Schürmann, and Julius L. Shaneson. Equivariant toric geometry and Euler-Maclaurin formulae. arXiv e-prints, March 2023. arXiv:2303.16785.

    $\bullet $ Arthur Bartels and Wolfgang Lück. On the algebraic $\mathrm K$-theory of Hecke algebras. In Mathematics Going Forward, Lecture Notes in Mathematics, pages 241–277. Springer International Publishing, January 2023. doi:10.1007/978-3-031-12244-6_19.

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    further publications

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Cohomology theories in arithmetic geometry

© MM/vl

Investigators: Deninger, Hartl, Hellmann, Lourenço, Nikolaus, Schürmann, Viehmann

In this unit we investigate cohomology groups in arithmetic geometry. The theories that are studied include coherent, étale, prismatic and topological periodic homology and are used to gain insights about various arithmetic schemes, including local and global Shimura varieties important in the Langlands programme and schemes that show up in studying generalised zeta functions, specifically in Deninger’s programme.

Algebraic K-theory and stable ∞-categories

© MM/vl

Investigators: Bartels, Hellmann, Hille, Mann, Nikolaus, Ramzi, Zhao

Throughout this unit, we focus on algebraic K-theory, which was originally defined for rings and schemes, but the modern approach views it as an invariant of stable $\infty$-categories which has led to many structural breakthroughs. This unit combines computational questions about K-theory with structural properties for stable $\infty$-categories. The applications concern algebraic and geometric topology, categorical aspects of the Langlands programme as well as homotopy theoretic questions.

Topological K-theory and group cohomology

© MM/vl

Investigators: Brück, Cuntz, Ebert, Joachim, Kramer, Nikolaus, Sroka, Winter, Zeidler

This unit deals with topological K-theory, which was originally defined for topological spaces but can, in its most general form, be viewed as an invariant of operator algebras and thereby used to gain insights into the structure of operator algebras. Through the use of topological methods and considerations, this is related to group cohomology, which we also study in this unit, specifically for arithmetic groups.