MA3F1 Introduction to topology
Term I 2017-2018
||Date of Monday
||Pages in Hatcher
|| 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.
||There is no lecture on Monday, and no support classes this week.
Questions asked by students on 2017-10-04.
|| Pairs of spaces, pointed spaces. Retracts, deformation
retracts. No retract theorem. Straight-line 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
Questions asked by a student on 2017-10-11.
|| 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
Questions asked by students on 2017-10-18.
|| \(\Phi\) is an isomorphism. Covering maps have the (unique)
homotopy lifting property.
||29 - 31, 60
||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
||31 - 34, 36
|| Borsuk-Ulam 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
Questions asked by a student on 2017-11-07.
|| Free products, words, empty, reduced. Proof of
associativity. Hawaiian earring. Statement of Seifert-van
Kampen. \( \pi_1(S^1 \vee S^1) = \ZZ \ast \ZZ\). First half of
the proof of SvK: factorisations.
||43 - 45
Question asked by a student on 2017-11-13.
|| 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
|| 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 two-complexes, algorithm to
|| 519 - 523, 50 - 52, 97 (only first paragraph)
|| Classification of surfaces. Construction of universal covers.
||63 - 70