Graduate Topology Seminar

Held semester 2, 2018

References:

[D-K] J.F Davis and P.Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics, 35.  American Mathematical Society, Providence, RI, 2001. http://www.indiana.edu/~lniat/book.pdf
[H] A. Hatcher, Algebraic topology, Cambridge University Press 2002 https://www.math.cornell.edu/~hatcher/AT/ATpage.html
[H2] A. Hatcher, Chapter 5 – Spectral Sequences, Still in progress, http://pi.math.cornell.edu/~hatcher/AT/SSpage.html
[H3] A. Hatcher, Vector Bundles and K-Theory, Still in progress, http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html
[Hu] D. Husemoller, Fibre bundles, Third edition, GTM 20, Springer-Verlag, New York, 1994
[M] J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, The University of Chicago Press, 1999
[M-S] J. Milnor and Stasheff, Characteristic classes, Ann. of Math. Studies, No. 76, Princeton University Press, Princeton, N. J., University of Tokyo Press, Tokyo, 1974
[S] R. M. Switzer, Algebraic topology – homotopy and homology, Classics in Mathematics, reprint of the 1975 original, Springer-Verlag, 2002

Lecture 1: Introduction to cohomology I

Dualise for cohomology; examples of cohomology theories: singular, de Rham
Speaker: Ethan Armitage
Tutor: Marcy Robertson
Reference: [H, Ch 3.1] and [D-K, Ch 2.6]
Time: 12:30 – 1:30pm, Friday 27/07 – Note the change of time
Notes

Lectures 2 and 3: Introduction to cohomology II

Mayer-Vietoris for cohomology; universal coefficient theorem; examples
Speaker: Michelle Strumila
Tutor: Diarmuid Crowley
Reference: [H, Ch 3.2], [D-K Ch 2.6]
Time: 1-3pm, 02/08 (week 2)

Lecture 4: Cup products I

Eilenberg-Zilber theorem; Alexander Whitney map
Speaker: Patrick Elliot
Tutor: Xing Gu
Reference: [D-K]
Time: 1-2pm, 09/08 (week 3)

Research talk 1: Infinity modular operads and the Teichmuller tower

The goal of Grothendieck-Teichmuller theory is to study the absolute galois group of the rational numbers via faithful actions on certain geometric objects. In this talk we define the Teichmuller tower and outline its use in studying the absolute galois group of the rationals. We then give the definition of an infinity modular operad and explain their potential for modelling the Teichmuller tower and end with an explicit example.

Speaker: Ethan Armitage
Time: 2-3pm, 09/08 (week 3)

Lecture 5: Cup products II

Acyclic Models, Kunneth formula
Speaker: Adam Wood
Tutor: Xing Gu
Time: 1-2pm, 16/08 (week 4)
Reference: [S, 13.24 – 13.28]
Notes

Exercise session (Lecture 6)

Tutor: Xing Gu
Time: 2-3pm, 16/08 (week 4)
Notes

Lecture 7: Poincare-Lefschetz duality I

Statement and proof
Speaker: Finn McGlade
Tutor: Huijun Yang
Time: 1-2pm, 23/08 (week 5)
Reference: [M, Ch 20], [H]
Notes

Research talk 2: Infinity cyclic operads and asteroidal sets

Operads are a multi-input version of categories, while dagger categories are ones where the morphisms have no sense of direction.  This talk will begin with an introduction to cyclic operads, which extend categories in both these directions.  We will then explore asteroidal sets, the cyclic analogue of dendroidal sets, in order to work with infinity cyclic operads.

Speaker: Michelle Strumila
Time: 2-3pm, 23/08 (week 5)
Notes

Lecture 8: Poincare-Lefschetz duality II

Completion of proof and examples
Speaker: Tom Dove
Tutor: Huijun Yang
Time: 1-2pm, 30/08 (week 6)
Reference: [M, Ch 21], [H]

Lecture 9: Serre spectral sequence I

Summary statement, proof, and examples
Speaker: Songqi Han
Tutor: TriThang Tran
Reference: [H2]
Time: 2-3pm, 30/08 (week 6)
Notes

Lecture 10: Serre spectral sequence II

Speaker: Songqi Han
Tutor: TriThang Tran
Reference: [H2]
Time: 1-2pm, 6/09 (week 7)
Notes are as above

Research talk 3

Speaker: Patrick Elliott
Time: 2-3pm, 6/09 (week 7)

Lecture 11: The cohomology ring H*(CPn)

Computation, relation to intersection of cycles and Poincar ́e duality
Speaker: Blake Sims
Tutor: Iva Halacheva
Reference:
Time: 1-2pm, 13/09 (week 8)

Lecture 12: Vector bundles and classifying spaces

Pull-back, universal bundles and Grassmannians
Speaker: Tom Dove
Tutor: Diarmuid Crowley
Reference: [M-S, Sections 2 and 5]
Time: 2-3pm, 13/09 (week 8)
Notes

Lecture 13: Classifying spaces and Characteristic classes

Definition of universal principal G-bundles and characteristic classes
Speaker: Blake Sims
Tutor: Marcy Robertson
Reference: [Hu, 4.1-4.7]
Time: 1-2pm, 20/09 (week 9)

Research talk 4

Speaker: Csaba Nagy
Time: 2-3pm, 20/09 (week 9)
Notes

Mid-semester Break – no seminar

Lecture 14: The Euler class and the Thom isomorphism

Definitions, sketches of proofs, examples
Speaker: Marcy Robertson
Reference: [M-S, Sections 9 and 10]
Time: 1-2pm, 4/10 (week 10)

Lecture 15: Poincare-Hopf theorem

Statement, proof, applications
Speaker: Blake Sims
Tutor: Diarmuid Crowley
Reference: [M, Ch. 5]
Time: 2-3pm, 4/10 (week 10)

Lecture 16: Chern classes

Definition and basic properties
Speaker: Ethan Armitage
Tutor: Diarmuid Crowley
Reference: [M-S, Ch. 14, pp. 155-173]
Time: 1-2pm, 11/10 (week 11)

Research talk 5 – Smooth parametrised Whitehead torsion and the A-theory presheaf

Dwyer, Weiss and Williams constructed a parametrised Whitehead torsion for smooth compact manifold bundles, which is an obstruction for a fibrewise bundle homotopy equivalence to be fibrewise homotopy equivalent to a bundle diffeomorphism. In this talk, I will describe the construction of the Becker-Gottlieb transfer and its equivalent construction by Dwyer, Weiss and Williams. Then I will briefly review their construction of parametrised Whitehead torsion and construct an A-theory presheaf.

Speaker: Songqi Han
Time: 2-3pm, 11/10 (week 11)
Notes

Lecture 17: An Introduction to Topological K-Theory

Definitions, basic properties, and the statement of Bott periodicity
Speaker: Tom Dove
Tutor: Iva Halaceva
Reference: [H3, 2.1 and 2.2]
Time: 1-2pm, 18/10 (week 12)

Lecture 18: Chern character

Definition, properties, complex bundles over spheres, division algebras
Speaker: Huijun Yang
Reference: [Hu, 20.9]
Time: 2-3pm, 18/10 (week 12)