MA3F1 Introduction to topology
Term I 20142015

Schedule
Week 
Date of Monday 
Topics 
Pages in Hatcher 
Example sheet 
Comments 
1 
Sep. 29 
Introduction. Topological spaces, products, subspaces,
quotients. Homeomorphisms. Invariance of domain. Peano curves.
Homotopic functions, homotopy equivalent spaces. 
1  3 
One 

2 
Oct. 6 
Homeomorphism and homotopy invariants. The questions of
topology. Straight line homotopy. Contractible spaces. Paths,
basepoints, loops, and concatenation. Based homotopies. The
fundamental group. \(\pi_1(\RR^n) = 1\). 
4, 25  27 
Two 

3 
Oct. 13 
Change of basepoint. Introduction to \(\pi_1(S^1)\).
Covering maps and spaces, degree of a cover, isomorphism of
covers, deck groups. Examples. 
28, 56  58, 67, 70 
Three 

4 
Oct. 20 
\(\Phi\) is a homomorphism. \(\Phi\) is an isomorphism.
Homotopy lifting property. 
29  31, 60 
Four 

5 
Oct. 27 
Gluing lemma. Finish homotopy lifting property. Applications
of \(\pi_1(S^1)\). Retractions, deformation retractions, induced
homomorphisms, "functoriality". Permutation representation of
fundamental group of the base space. 
31  34, 36, 68  70 
Five 

6 
Nov. 3 
No retraction from ball to sphere, the Brouwer fixed point
theorem. The fundamental groups of spheres, of products.
\(\pi_1\) is a homotopy invariant. Covers induce injections on
\(\pi_1\), degree equals index. Free products, reduced
words. 
31  32, 35, 37, 41  42, 61 
Six 

7 
Nov. 10 
Classification of orientable surfaces, connect sum. \( T \# P
= 3P \). Statement of Seifertvan Kampen. \( \pi_1(S^1 \vee S^1)
= \ZZ \ast \ZZ\). First half of the proof of SvK:
factorizations. 
43  45 
Seven 

8 
Nov. 17 
Second half of the proof of SvK: reductions, expansions,
exchanges. Cells, boundaries, attaching maps, CW complexes,
subcomplexes, graphs, trees. 
5, 7, 45  46, 519  523 
Eight 
The eighth example sheet is closely related to pages 83  87
in Hatcher.

9 
Nov. 24 
\(\pi_1\) of CW complexes using SvK. Presentations of groups,
examples. Presentations of \(\pi_1\) of twocomplexes,
examples. 
50  52, 97 
Nine 
Posted a new version of ninth example sheet on 20141126.

10 
Dec. 1 
Attaching threecells, fundamental group determined by
twoskeleton. Isomorphic covers, Galois correspondence. Local
properties of topological spaces, construction of universal
covers, examples. 
62  68 
Ten 



