MA3F1 Introduction to topology
Term I 20152016

Schedule
Week 
Date of Monday 
Topics 
Pages in Hatcher 
Example sheet 
Comments 
1 
Oct. 5 
Introduction. Topological spaces, products, subspaces,
quotients, disjoint unions. Homeomorphism, foundational problem.
Invariance of domain. Topological invariants. Homotopies,
homotopy equivalence. 
1  3 
One 
There are no support classes this week. 
2 
Oct. 12 
Gluing lemma. Straightline homotopy. Contractible spaces.
Paths, concatenation, basepoints, loops. Based homotopies. The
fundamental group. \(\pi_1(\RR^n) \isom 1\). Change of
basepoint.

4, 25  28 
Two 
Lectures by Prof. David Mond.

3 
Oct. 19 
Introduction to \(\pi_1(S^1)\) and winding number. Covering
maps, covering spaces. Degree of a cover, isomorphism and
automorphism of covers, deck groups. Retractions, deformation
retractions. Lifting in general, lifting for the circle.

29, 56  60, 67, 70 
Three 

4 
Oct. 26 
Homotopies descend, concatenation descends. \(\Phi\) is a
homomorphism. \(\Phi\) is an isomorphism. Gluing lemma, homotopy
lifting property, generalizes path lifting property. 
29  31, 60 
Four 
Questions asked by
students.

5 
Nov. 2 
Finish homotopy lifting property. Induced homomorphisms,
functorality. Application to retractions, deformation
retractions. \(\pi_1\) is a homotopy invariant. No retraction
theorem, the Brouwer fixed point theorem, variants. 
31  34, 36 
Five 
Questions asked by
students. On the assignment it may be useful to know that
\(\pi_1(X \cross Y) \isom \pi_1(X) \times \pi_1(Y)\).

6 
Nov. 9 
Even versus odd, BorsukUlam theorem, variants. Covers
induce injections on \(\pi_1\), degree equals index. The
fundamental groups of spheres, of cartesian products. Motavation
for free groups. 
31  32, 35, 41  42, 61 
Six 
Questions asked by
students.

7 
Nov. 16 
Free products, words, empty, reduced. Proof of
associativity. Hawaiian earring. 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 
Questions asked by
students.

8 
Nov. 23 
Second half of the proof of SvK: reductions, expansions,
exchanges. Cells, boundaries, attaching maps, CW complexes, weak
topology, subcomplexes, finite, finite dimensional, graphs, trees.
Propositions A.1, A.3, A.4, A.5. 
5  7, 45  46, 519  523 
Eight 

9 
Nov. 30 
\(\pi_1\) of CW complexes using SvK. Presentations of
groups. Presentations of \(\pi_1\) of twocomplexes, algorithm to
compute presentations. 
50  52, 97 (only first paragraph) 
Nine 
Questions asked by
students.

10 
Dec. 7 
Classification of surfaces. Isomorphic covers, Galois
correspondence. Construction of universal covers. 
63  70 
Ten
Eleven 
Questions asked by
students.



