# Algebraic Topology II Spring 2020

Lecturer
Alessandro Sisto
Coordinator
Paula Truöl

Classes are held in ML E 12 on Wednesdays 10-12, and in HG G 3 on Fridays 13-15.

## Content

This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality.

Course Catalogue

## Prerequisites

General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.

## Lecture Summaries

Day Pages Summary
Week 1 (Feb. 19 and 21) Hatcher, page 190 to corollary 3.3 (except the proof of Lemma 3.1) Moral introduction to cohomology, cohomology of a chain-complex, universal coefficient theorem
Feb. 26 Lemma 3.1, pages 197-198 Ext doesn't depend on the resolution, definition of cohomology of a space
Feb. 28 Pages 199-204 Basic properties of cohomology, dual to those for homology
March 4 Section 3.2 until Example 3.7 excluded, plus Proposition 3.10. (More discussion on pages 185-186.)

Definition and basic properties of cup products

March 6 Examples 3.7 and and 3.8 (both more general than discussed in class)

Note: We will develop tools to make those computations in different ways

Examples of interesting cup products: projective plane and torus

March 11 Pages 210-212

March 13 Pages 213-216

Examples of cohomology rings, cross product, tensor products, statement of the Kuenneth formula

March 18 Pages 216-218

Outline of proof of the Kuenneth formula

March 20 Pages 209-210, Theorem 3.18 (statement), and pages 220-222.

Another version of the relative cup product, cohomology of projective spaces

March 25 Exercise Class 1 Notes on Exercise Class 1 and Recording of Exercise Class 1 on Zoom
March 27 Example 3.20, Theorem 3.21

If R^n is a division algebra, then n is a power of 2

April 1

The complex Hopf map (not needed for the exam)

April 3 Pages 224-225

James reduced product (not needed for the exam)

April 8 Exercise Class 2 Notes on Exercise Class 2 and Recording of Exercise Class 2 on Zoom
April 22 Pages 231 and 233-234

Local orientations and orientations, orientation cover (statement only)

April 24 Pages 234-236

Orientation cover (proof), R-orientability, structure of top dimensional homology (statement), fundamental class

April 29 Pages 236-237

Structure of top dimensional homology (proof)

May 6 Pages 239-245

Cap product, Poincare duality, cohomology with compact support

May 8 Pages 246-249

Proof of Poincare' duality (not part of the exam)

May 13 Pages 254-256, and Corollary 3.28+Proposition 3.29

Alexander duality and related results (not needed for the exam)

May 15 Pages 525-527 and 255

Euclidean neighborhood retracts (not needed for the exam)

May 20 Exercise Class 3 Notes on Exercise Class 3 and Recording of Exercise Class 3 on Zoom
May 27 Pages 87-91

Aspherical spaces, definition of group homology and cohomology (not needed for the exam)

May 29

Central extensions (not needed for the exam)

## Exercises

The new exercises will be posted here.

Exercise sheet Solutions Notes
Exercise sheet 1 Solutions 1
Exercise sheet 2 Solutions 2 Small typo in the solution of exercise 4 corrected
Exercise sheet 3 Solutions 3 Typos in exercise 3&4 corrected
Exercise sheet 4 Solutions 4 Added a sentence in the solution of exercise 4
Exercise sheet 5 Solutions 5 Added a paragraph in the solution of exercise 3

## Feedback form

You can send any comments/suggestions/complaints using this anonymous form.

## Literature

• A. Hatcher, "Algebraic Topology", Cambridge University Press, Cambridge, 2002. See also here.
• G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.
• E. Spanier, "Algebraic Topology", Springer-Verlag.