GRK 1150:  Homotopy and Cohomology

• Subject Area: Mathematics
• Term: from 2005 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 796242

• Final Report Year: 2016

Final Report Abstract

Die Topologie ist ein Teilgebiet der Mathematik, welches die qualitativen Eigenschaften und nicht die quantitativen Eigenschaften von geometrischen Objekten wie Polyedern oder Mannigfaltigkeiten (von beliebiger Dimension) untersucht. Quantitative Eigenschaften wären etwa Kurvenlängen, Flächeninhalte, Volumen, Winkel und Krümmung; im Gegensatz dazu sind Zusammenhang oder Verschlingungszustände qualitative Eigenschaften, die unter kontinuierlichen Deformationen (sog. Homotopien) der Gegenstände invariant bleiben. Diese Eigenschaften werden durch Invarianten gemessen und sind im einfachsten Fall nur Zahlen, z.B. die berühmte Euler-Charakteristik χ(M ) = b0 (M ) − b1 (M ) + b2 (M ) − . . . + (−1)n bn (M ) für einen n-dimensionalen Polyeder M, wobei bk(M) die Anzahl der k-dimensionalen (und einfachen) Stücke von M bezeichnet. Meistens sind diese Invarianten aber wesentlich komplizierter, nämlich Elemente in einer Kohomologie h*(M) (oder auch Homologie) der geometrischen Objekte. Solche Kohomologien h* gibt es viele und manchmal wird für ein ungelöstes topologisches Problem sogar erst eine dazu passende Invariante in einer neuen Kohomologietheorie konstruiert, um das Problem zu studieren und vielleicht zu lösen. Die Klassifikation z.B. aller Mannigfaltigkeiten einer bestimmten Dimension oder mit bestimmten Eigenschaften ist ein zentrales Thema der Topologie. Wo Klassifikation zu schwierig ist, kann man wenigsten fragen, ob zwei Mannigfaltigkeiten Mi,M2 ineinander deformierbar sind oder nicht: weiß man von einer Invarianten I, dass I(M1 ) = I(M2 ) gilt, so sind M1 und M2 nicht ineinander deformierbar. Zur Beantwortung solcher Fragen ist die ständige Weiterentwicklung und Verfeinerung der Invarianten wichtig. Im GRK 1150 Homotopie und Kohomologie sind u.a. folgende Fragen angegangen worden: • Homotopieklassifikation von Mannigfaltigkeiten (Vermutungen von Borel, Atiyah, Farrell-Jones); • Knoten (und Verallgemeinerungen) und ihre Klassifikation; • Kohomologie von Gruppen und ihrer klassifizierenden Räume; • Quantenfeldtheorien und Stringtheorie; • symmetrische Spektren und globale äquivariante Homotopietheorie; • Flächen und die Kohomologie ihrer Modulräume. In etlichen Fragen sind hierbei durch die Dissertationen bemerkenswerte Ergebnisse erzielt worden, wie man den Berichten der Doktoranden entnehmen kann.

Publications

• . A classification of 5-dimensional manifolds, souls of codimension two and non-diffeomorphic pairs
  Ottenburger, Sadeeb
  
• A diagram algebra for Soergel modules corresponding to smooth Schubert varieties. In: Trans. Amer. Math. Soc.
  Sartori, Antonio
  
• A geomertic description of the Atiyah-Hirzebruch Spectral Sequence for B-Bordism
  Tene, Haggai
  
• Categorification of tensor powers of the vector representation of Uq(gl(1|1)). In Selecta Math.
  Sartori, Antonio
  
• Comparing combinatorial models of moduli space and their compactifications
  Egas Santander, Daniela, and Kupers, Sander
  
• Comparing graph modes of moduli space
  Egas Santander, Daniela
  
• Derived induction and restriction theory
  Noel, Justin, Akhil Mathew and Niko Naumann
  
• Derived smooth manifolds: A simplicial approach
  Noel, Justin, Dennis Borisov
  
• Discrete Morse Theory and a Reformulation of the K(π, 1)-conjecture
  Ozornova, Viktorya
  
• Expansion principles for topological automorphic forms
  Thyssen, Sebastian
  
• Factorability, String Rewriting and Discrete Morse Theory
  Ozornova, Viktorya, Alexander Heß
  
• Generalized Snaith splittings
  Wang, Juan
  
• Generalized Torelli Groups
  Siegmund, Marc
  
• Hecke Operators for Non-Congruence Subgroups of Bianchi Groups
  Hamzeh-Zarghani, Saeid
  
• Hilbert stratifolds and a Quillen type geometric description of cohomology for Hilbert manifolds
  Tene, Haggai, M. Kreck
  
• K(1)-local complex E∞-orientations
  Allenberg, Jan (geb. Möllers)
  
• Obstruction theory for operadic algebras
  Dimitrova, Boryana
  
• On e- and f-invaiants of complex reframings
  Beckers, Achim
  
• On K(1)-local SU-bordism
  Reeker, Holger
  
• On rigidity of the ring spectra Pm S(p) and ko
  Hutschenreuter, Katja
  
• On the geometry of the f -invariant
  von Bodecker, Hanno
  
• On the group cohomology of the semi-direct product Zn x Z/m and a conjecture of Adem-Ge-Pan-Petrosyan
  Langer, Martin, Wolfgang Lück
  
• Products in equivariant homology
  Tene, Haggai, S. Kaji
  
• Quasicategories of Frames of Cofibration Categories
  Szumilo, Karol, Chris Kapulkin
  
• Rigidity in equivariant stable homotopy theory. Algebraic and Geometric Topology
  Patchkoria, Irakli
  
• Rigidity in equivariant stable homotopy theory. In: Algebraic and Geometric Topology
  Patchkoria, Irakli
  
• Rigidity in equivariant stable homotopy theory. In: Algebraic and Geometric Topology
  Stojanoska, Vesna
  
• Secondary multiplication in Tate cohomology of generalized quaternion groups
  Langer, Martin
  
• Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds
  Ottenburger, Sadeeb
  
• Some applications of the theory of formal groups to algebraic topology. PhD Thesis (University of Chicago)
  Noel, Justin
  
• Some geometric equivariant cohomology theories
  Tene, Haggai
  
• Topological K-theory of the group C*-algebra of a semi-direct product Zn x Z/m for a free conjugation action
  Langer, Martin, Wolfgang Lück
  
• Two Models for the Homotopy Theory of Cocomplete Homotopy Theories
  Szumilo, Karol
  
• Universality of multiplicative infinite loop space machines. In: Algebraic & Geometric Topology
  Groth, Moritz with Gepner and Nikolaus
  
• Über den Ext-Funktor auf diskontinuierlichen Räumen
  Deniz, Ferit
  
• Factorability, String Rewriting and Discrete Morse Theory
  Heß, Alexander, Viktoriya Ozornova
  
• Nilpotence and descent in equivariant stable homotopy theory
  Noel, Justin, Akhil Mathew and Niko Naumann
  
• Air-Breathing Laminar Flow-Based Direct Methanol Fuel Cell with Alkaline Electrolyte. Electrochem. Solid-State Lett. 9 (5), pp. A252-A256 (2006)
  Ranga S. J., Spendelow J.S., Natarajan D., Markoski L. J. and Kenis P.J.A.
  
• Popaths and Holinks. Journal of Homotopy and Related Structures, vol. 1 (2006), 1–9
  Miller, David
  
• Rigidity and exotic models for k-local spectra. Geometry & Topology 11 (2007), 1855 - 1886
  Roitzheim, Constanze
  
• On the algebraic classification of K-local Spectra. Homology, Homotopy, and Appl. 10 (2008), 389 - 412
  Roitzheim, Constanze
  
• A diffeomorphism classification of 5- and 7-dimensional non-simply-connected homogeneous spaces
  Ottenburger, Sadeeb
  
• On the notion of order in the stable module category
  Langer, Martin
  
• Ruthenium clusterlike chalcogenide as a methanol tolerant cathode catalyst in air-breathing laminar flow fuel cells. Electrochimica Acta, 54 (18) pp. 4384-4388 (2009)
  Egas Santander, Daniela, Devin T. W., Ranga S. J., Alonso-Vante N. and Kenis P.J.A.
  
• For Complex Orientations Preserving Power Operations, p-typicality is Atypical. Topology and its Applications, October 2010
  Noel, Justin, Niles Johnson
  
• Homotopy theory of s-bimodules, naive ring spectra and stable model categories
  Weiner, Arne
  
• Controlled algebra for simplicial rings and the algebraic K-theory assembly map. Dissertation (2011), Universität Düsseldorf
  Ullmann, Marc
  
• Spectral Sequences in String Topology. Algebraic & Geometric Topology 11 (2011), 2829–2860
  Meier, Lennart
  
• Stratifolds and Equivariant Cohomology Theories
  Tene, Haggai
  
• A graphical calculus for 2-block Spaltenstein varieties. Glasg. Math. J. 54 (2012), 449–477
  Schäfer, Gisa
  
• Cubical approach to derived functors. Homology, Homotopy and Appl. 14 (2012), 133 – 158
  Patchkoria, Irakli
  
• Cubical approach to derived functors. Homology, Homotopy and Appl. 14 (2012), 133 – 158
  Patchkoria, Irakli
  
• Cubical approach to derived functors. Homology, Homotopy and Appl. 14 (2012), 133 – 158
  Stojanoska, Vesna
  
• On the algebraic classification of module spectra. Algebraic and Geometric Topology 12, (2012), 2329 – 2388
  Patchkoria, Irakli
  
• On the algebraic classification of module spectra. Algebraic and Geometric Topology 12, (2012), 2329 – 2388
  Patchkoria, Irakli
  
• On the algebraic classification of module spectra. Algebraic and Geometric Topology 12, (2012), 2329 – 2388
  Stojanoska, Vesna
  
• On the Product in Negative Tate Cohomology for finite groups, Algebraic and Geometric Topology, 12, 2012
  Tene, Haggai
  
• Secondary multiplication in Tate cohomology of certain p-groups. Algebr. Geom. Topol. 12 (2012) 829-865
  Langer, Martin
  
• Derivators, pointed derivators and stable derivators. Algebraic & Geometric Topology 13 (2013), pp- 313-374
  Groth, Moritz
  
• Strongly stratified homotopy theory. Trans. Amer. Math. Soc. 365 (2013), 4933-4962
  Miller, David
  
• Controlled Algebra for Simplicial Rings and Algebraic K-theory
  Ullmann, Marc
  
• Fibrancy of Partial Model Categories. Homology, Homotopy Appl.
  Meier, Lennart, Viktoriya Ozornova
  
• Fibrancy of Partial Model Categories. In: Homology, Homotopy Appl.
  Ozornova, Viktorya, Lennart Meier
  
• H∞ ≠ E∞. Contemporary Mathematics, vol. 617, Amer. Math. Soc., 2014
  Noel, Justin
  
• Lifting homotopy T -algebra maps to strict maps. Advances in Mathematics, vol. 264(20) (2014), pp. 593–645
  Noel, Justin, Niles Johnson
  
• Mayer–Vietoris sequences in stable derivators. Homology, Homotopy and Applications 16 (2014), pp. 365-394
  Groth, Moritz with Ponto and Shulman
  
• The additivity of traces in monoidal derivators. Journal of K-Theory 14 (2014), pp. 422-494
  Groth, Moritz with Ponto and Shulman
  
• The degenerate affine walled Brauer algebra. J. Algebra 417 (2014), 198–233
  Sartori, Antonio
  
• The T-algebra spectral sequence: Comparisons and applications. Algebraic and Geometric Topology, 14-6 (2014), 3395–3417
  Noel, Justin
  
• The first differential of the functor ’Algebraic K-theory of Spaces’
  Ullmann, Marc
  
• A model structure on GCat. In: Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics vol. 641 (2015), S. 123 – 134
  Ozornova, Viktorya, Anna Marie Bohmann, Kristen Mazur, Angélica M. Osorno, Kate Ponto, Carolyn Yarnall
  
• All about T mf1(3)
  Meier, Lennart, Michael Hill
  
• Affineness and Chromatic Homotopy Theory. In: J Topology (2015) 8 (2): 476-528
  Meier, Lennart, Akhil Mathew
  
• Categorification of tensor product representations of slk and category O. J. Algebra 428 (2015), 256—291
  Sartori, Antonio, C. Stroppel
  
• Fibration Categories are Fibrant Relative Categories
  Meier, Lennart
  
• Nilpotence in the symplectic bordism ring. Contemporary Mathematics, Proceedings of the Conference on Manifolds, K-Theory, and Related Topics, Amer. Math. Soc.
  Noel, Justin
  
• On a nilpotence conjecture of J. P. May. Journal of Topology, 8, Issue 4, 1 December 2015, Pages 917–932
  Akhil Mathew, Niko Naumann, Justin Noel
  
• On auto-equivalences of the (∞, 1)-category of ∞-operads. Mathematische Zeitschrift 281 (2015), 807-848
  Groth, Moritz with Ara and Gutierrez
  
• The Alexander polynomial as quantum invariant of links. Ark. Mat. 53, Issue 1 (2015), Pages 177-202
  Sartori, Antonio
  
• Vector Bundles over the Moduli Stack of Elliptic Curves. J. Algebra 428 (2015), 425-456
  Meier, Lennart
  
• Tilting theory of trees via stable homotopy theory. Journal of Pure and Applied Algebra 220 (2016), 2324-2363
  Groth, Moritz with Stovicek