Math 8360: Topics in Topology
Rational Homotopy Theory

COURSE SYLLABUS

Fall 2021

INSTRUCTOR: Sasha Voronov

OFFICE: VinH 324

PHONE: (612) 624-0355

E-MAIL ADDRESS: voronov@umn.edu. You are welcome to use e-mail to send questions to me.

INTERNET: All class announcements and assignments will be posted on the class homepage http://www.math.umn.edu/~voronov/8360/index.html and NOT handed out in class.

OFFICE HOURS (tentative): Mon 1:25-2:15, Wed 2:30-3:20, Thu (on Zoom) 2:30-3:20, or by appointment.

TEXT:  I do not recommend any particular textbook, even though they exist: there are a few monographs, lecture notes, and surveys on the subject. I will mostly follow Julian Holstein's online notes, which is a modern exposition of the short monograph/long research paper [BG76]. However, when discussing Quillen minimal models, I will take a novel, arguably more natural path: instead of free graded Lie algebras, I will be consistently using L_∞-algebras in the spirit of [BFM13] and [Kon03]. The books [FHT01, FHT13, FOT08] have many applications and are excellent as reference books. The monograph [GM13] takes a somewhat different approach and could be a useful alternative.

• [BFM13] Buijs, U.; Félix, Y.; Murillo, A. L_∞ rational homotopy of mapping spaces. Rev. Mat. Complut. 26 (2013), no. 2, 573-588, arXiv:1209.4756 [math.AT]
• [BG76] Bousfield, A. K.; Gugenheim, V. K. A. M. On PL de Rham theory and rational homotopy type. Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94 pp.
• [FHT01] Félix, Y.; Halperin, S.; Thomas, J.-C. Rational homotopy theory. Graduate Texts in Mathematics, 205. Springer-Verlag, New York, 2001. xxxiv+535 pp.
• [FHT13] F élix, Y.; Halperin, S.; Thomas, J.-C. Rational homotopy theory. II. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. xxxvi+412 pp.
• [FOT08] Félix, Y.; Oprea, J.; Tanré, D. Algebraic models in geometry. Oxford Graduate Texts in Mathematics, 17. Oxford University Press, Oxford, 2008. xxii+460 pp.
• [GM13] Griffiths, P.; Morgan, J. Rational homotopy theory and differential forms. Second edition. Progress in Mathematics, 16. Springer, New York, 2013. xii+224 pp.
• [Kon03] Kontsevich, M. Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), no. 3, 157-216.

CONTENT: This is a one-semester course devoted to rational homotopy theory. I plan to cover dg-commutative algebras, model categories, simplicial sets, polynomial de Rham algebra, spectral sequences, Quillen models (the L_∞ version), and an application to Mysterious Duality in string theory, time permitting.

PREREQUISITES:The expected background is familiarity with basic notions of algebraic topology: such as homology and homotopy groups. More advanced topics of algebraic topology, such as simplicial sets, model categories, L_∞-algebras, also known as strongly homotopy Lie algebras, and spectral sequences will be introduced along the way. Thus, even if you are not interested in rational homotopy theory per se, you may benefit from learning these important tools, which are used throughout algebraic topology.

GRADING: Since understanding the material, keeping up with course and reviewing your notes will already present certain challenge, no homework or tests will be given. If you make effort to make sure I remember you by the end of the term, you will get an A.

IMPORTANT DATES:

September 8: First class meeting.

November 25-26: Thanksgiving Break.

December 15: Last class meeting.

Page created on 09/10/2021. Office hours updated on 09/20/21.