# Algebraic Topology I Autumn 2019

Lecturer
Alessandro Sisto
Coordinator
Davide Spriano

Classes are held in HG D7.1, on Wednesday 10-12, Friday 13-15.

Note! The first three lectures (18, 20, 25 Sept) are held in HG E 3.

## Content

This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms.

## Prerequisites

You should know the basics of point-set topology.

Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology").

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

Some (elementary) group theory and algebra will also be needed.

## Lecture Summaries

Day pages summary
Week 1 (18-20 Sept) Bredon 168-171, Hatcher 103-104-108 (bottom) -109 (top) Definitions of simplices, chains, cycles, boundaries, homology Geometric intuition (Delta-complexes, orientations)
25 Sept Bredon 171-173 Homology of a point, H_0(X) is determined by path-components, Hurewicz's Theorem: statement and beginning of proof
27 Sept Bredon 173-175 End of proof of Hurewicz's Theorem.
2 Oct Bredon 176-181 Induced maps in homology, chain complexes, chain maps, long exact sequence of homology groups (statement), homology with coefficients, relative homology, some notions of category theory. Note: category theory is not required for the exam, and was mentioned only to provide context
9 Oct Bredon 178-179 and 183-184 Proof of the theorem on long exact sequences in homology (Theorem 5.6), definition of homology theory, homology of spheres (statement)
11 Oct Bredon 184-186 and 194-195 Homology of spheres, reduced homology, Brouwer's fixed point theorem, definition of CW-complex
16 Oct Bredon 195-197 CW-complexes: examples, compact subspaces, (very) rough outline of homology computation
18 Oct Bredon 200-205 Computation of the homology of CW-complexes, definition of degree
23 Oct Bredon 187-188, 203-204 Clarifications on homology of CW-complexes, basic properties of degree, reflections have degree -1.
30 Oct Bredon 190-192 (relevant for exam: Proposition 7.3 and Theorem 7.4) Computation of degrees
1 Nov Bredon 204, 207-210, 215-216 Relative homology of CW-complexes (no proofs), cellular approximation theorem and induced maps (no proofs), Euler characteristic
6 Nov Example of computation of homology of CW complexes
8 Nov Bredon 219, Hatcher 111-113 Chain homotopies, proof of the homotopy axiom
13 Nov Bredon, 220-224 Cross products (no proofs), barycentric subdivisions
15 Nov Bredon, 224-226* *plus a more direct conclusion to the proof of excision than in Bredon. Barycentric subdivision is chain-homotopic to the identity, subdvision operator for general chains, proof of the excision axiom.
20 Nov Bredon, 227, 229-230 Homology with small simplices, Mayer-Vietoris sequence
22 Nov Bredon, 234-235 (see also 231-234), Dold (if you cannot access, try to connect from the ETH Wi-Fi) Homology of complements in R^n, Jordan curve theorem and other corollaries, invariance of domain and invariance of dimension.
27 Nov Bredon, 233-234, 236 Generalized Jordan curve theorem, Jordan-Brouwer separation theorem, statement of generalised Schoenflies theorem
29 Nov Bredon, 236-239 Proof of the generalised Schoenflies theorem (not needed for the exam*) *but very cool!
4 Dec Bredon, 245-248, 250-251 Simplicial complexes, simplicial homology, definition of simplicial map, barycentric subdivion
6 Dec Bredon, 251-253 Simplicial approximation theorem
11 Dec Bredon, 253-256 (minus ENR discussion) Homology with R or field coefficients, Lefschetz-Hopf fixed point theorem, sample applications (The content of this lecture is not needed for the exam).
13 Dec Bredon, 240-242 Borsuk-Ulam theorem (not needed for the exam)
18 Dec Exercises
20 Dec Final lecture: a discussion of some applications of algebraic topology (not needed for exam).

## Exercises

The new exercises will be posted here.

If you have questions regarding the exercises, please don't hesitate to contact Davide Spriano.

exercise sheet solutions notes
Exercise sheet 1 Solutions 1 Path connected -> path connected component. Also, allowed arbitrary unions
Exercise sheet 2 Solutions 2
Exercise sheet 3 Solutions 3
Exercise sheet 4 Solutions 4 Corrected definition of $$U$$ in Question 3, Solution of Question 4
Exercise sheet 5 Solutions 5 $$j_i \colon X_i \to X$$, corrected definition of splitting Question 3.
Exercise sheet 6 Solutions 6
Exercise sheet 7 Solutions 7 Added bonus part Question 3.
Exercise sheet 8 Solutions 8; PartII Corrected solution Ex 3.
Exercise sheet 9 Solutions 9; PartII
Exercise sheet 10 Solutions 10; PartII Expanded the details on computation of homology
Exercise sheet 11 Solutions 11
Exercise sheet 12 Solutions 12
Exercise sheet 13 Solutions 13
Mock Exam Solutions

## Feedback form

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

## Literature

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