MA3F1 Introduction to topology
Term I 20172018

Schedule
Week 
Date of Monday 
Topics 
Pages in Hatcher 
Example sheet 
Comments 
1 
Oct. 2 
Introduction and overview. Topological spaces, bases,
products, subspaces, examples. Homeomorphism, foundational
problem, invariance of domain, topological invariants. Disjoint
union, quotients, our friend the square, the Mobius band.

NA 
One 
There is no lecture on Monday, and no support classes this week.
Questions asked by students on 20171004. 
2 
Oct. 9 
Pairs of spaces, pointed spaces. Retracts, deformation
retracts. No retract theorem. Straightline homotopy, homotopy,
gluing lemma, homotopy equivalence. Contractible spaces, spheres
are not contractible. Paths, concatenation, homotopy rel
endpoints. Loops, the fundamental group. \(\pi_1(\RR^n) \isom
1\). Introduction to \(\pi_1(S^1)\) and winding number.

1  4, 25  27 
Two 
Questions asked by a student on 20171011. 
3 
Oct. 16 
Basepoints. Covering maps, covering spaces. Isomorphism of
covers, degree of a cover, deck groups. The rose with \(n\)
petals, its covers. Lifting. Homotopies descend, concatenations descend.
\(\Phi\) is a homomorphism.

28  29, 56  60, 67, 70 
Three 
Questions asked by students on 20171018. 
4 
Oct. 23 
\(\Phi\) is an isomorphism. Covering maps have the (unique)
homotopy lifting property. 
29  31, 60 
Four 

5 
Oct. 30 
Induced homomorphisms, functorality. Application to
retractions, deformation retractions. \(\pi_1\) is a homotopy
invariant. No retraction theorem, the Brouwer fixed point
theorem, variants. The fixed point property for a space. Even
versus odd. 
31  34, 36 
Five 

6 
Nov. 6 
BorsukUlam theorem. Covers induce injections on \(\pi_1\),
degree equals index. The fundamental groups of spheres, of
cartesian products. Motivation for free groups. 
31  32, 35, 41  42, 61 
Six 
Questions asked by a student on 20171107.

7 
Nov. 13 
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: factorisations. 
43  45 
Seven 
Question asked by a student on 20171113.

8 
Nov. 20 
Second half of the proof of SvK: reductions, expansions,
exchanges. SvK and \(\pi_1(\RP^2)\). Cells, boundaries,
attaching maps, CW complexes, weak topology, subcomplexes, finite,
finite dimensional, graphs, trees.

5  7, 45  46 
Eight 

9 
Nov. 27 
Propositions A.1, A.3, A.4, A.5. Local properties. Warsaw
circle. \(\pi_1\) of CW complexes using SvK. Presentations of
groups. Presentations of \(\pi_1\) of twocomplexes, algorithm to
compute presentations. 
519  523, 50  52, 97 (only first paragraph) 
Nine 

10 
Dec. 4 
Classification of surfaces. Construction of universal covers. 
63  70 
Ten
Eleven 



