Advanced Topics Course: Loday Constructions (Master)

	Advanced Topics Course: Loday constructions (Master), Winter term 2024/25
	Birgit Richter, email: birgit.richter at uni-hamburg.de
Plan:	The Loday construction is a device that takes a commutative algebra and a simplicial set and turns them into a simplicial commutative algebra. Its homotopy groups then give important invariants of both inputs. For the circle you actually don't have to assume commutativity and the Loday construction for \(S^1\) of an algebra \(A\) yields the Hochschild homology of \(A\). Other important examples are the Loday construction for higher dimensional spheres and torus homology. In the later part we will turn to an equivariant version of the Loday construction. Here the input is a finite simplicial \(G\)-set and a \(G\)-Tambara functor. This is an advanced topics course, so I will assume that you've had an algebraic topology course and some background in algebra. I will introduce simplicial sets.
When?	Monday, 12-14h, H5
	This is one of my research topics at the moment. If you plan to do a master thesis related to this lecture course, then please contact me as early as possible.
Exam:	The final exam for this course is an oral exam after the end of term.
Literature:	For background on simplicial sets you could use my book From Categories to Homotopy Theory, Peter May's book Simplicial Objects in Algebraic Topology (Chicago Lectures in Mathematics, The University of Chicago Press, 1993), Paul Goerss and Rick Jardine's book Simplicial Homotopy Theory (Modern Birkhäuser Classics, Springer 2009) or Peter Gabriel, M. Zisman's book Calculus of Fractions and Homotopy Theory (Softcover reprint of the original 1st ed. 1967, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 2012). For Hochschild homology you could use Chuck Weibel's book, An introduciton to homological algebra (Cambridge University Press) or Jean-Louis Loday's book, Cyclic Homology (Springer). Other references that I use: Donald W. Anderson, Chain functors and homology theories, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971), Lecture Notes in Mathematics 249, 1971, pp. 1--12. Gabriel Angelini-Knoll, Teena Gerhardt, Michael Hill, Real topological Hochschild homology via the norm and Real Witt vectors, arXiv:2111.06970. Yuri Berest, Ajay Ramadoss, Wai-Kit Yeung, Representation Homology of Topological Spaces, International Mathematics Research Notices, Volume 2022, Issue 6, March 2022, pp. 4093--4180, Irina Bobkova, Eva Höning, Ayelet Lindenstrauss, Kate Poirier, Birgit Richter and Inna Zakharevich, Splittings and calculational techniques for higher THH, Algebraic & Geometric Topology 19 (7) (2019), 3711--3753. Irina Bobkova, Ayelet Lindenstrauss, Kate Poirier, Birgit Richter and Inna Zakharevich, On the higher topological Hochschild homology of \(\mathbb{F}_p\) and commutative \(\mathbb{F}_p\)-group algebras, Women in Topology: Collaborations in Homotopy Theory. Contemporary Mathematics 641, AMS, 2015, 97--122. Aldrige K. Bousfield, Dan M. Kan, The core of a ring. J. Pure Appl. Algebra 2 (1972), 73--81. Albrecht Dold, René Thom, Quasifaserungen und unendliche symmetrische Produkte , Ann. Math. (2) 69 (1959), 239--281 Emanuele Dotto, Kristian Moi, Irakli Patchkoria, Sune Precht Reeh, Real topological Hochschild homology, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 1, 63--152. Bjørn Ian Dundas Andrea Tenti, Higher Hochschild homology is not a stable invariant. Math. Z. 290 (2018), no. 1-2, 145--154. Alice Hedenlund, Sarah Klanderman, Ayelet Lindenstrauss, Birgit Richter and Foling Zou, Loday constructions on twisted products and on tori, Topology Appl. 316 (2022), Paper No. 108103, 25 pp. Michael Hill, Kristin Mazur, An equivariant tensor product on Mackey functors, J. Pure Appl. Algebra 223 (2019), no. 12, 5310--5345. Ayelet Lindenstrauss, Birgit Richter, Reflexive homology and involutive Hochschild homology as equivariant Loday constructions, arXiv:2407.20082. Ayelet Lindenstrauss, Birgit Richter, Stability of Loday constructions, Homology, Homotopy and Applications, 24 (1), (2022), 401--425. Ayelet Lindenstrauss, Birgit Richter, Foling Zou, Loday constructions of Tambara functors, arXiv:2401.04216. Saunders Mac Lane, Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology, Annales scientifiques de l'É.N.S. 4e série, tome 33, no 2 (2000), p. 151--179. Neil Strickland, Tambara functors, arXiv:1205.2516. Daisuke Tambara, On multiplicative transfer, Comm. Algebra 21 (1993), no. 4, 1393--1420. Jacques Thévenaz, Peter Webb, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865--1961. Torleif Veen, Detecting periodic elements in higher topological Hochschild homology. Geom. Topol. 22 (2018), no. 2, 693--756.