MA3F1 Introduction to topology
Term I 2024-2025
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Schedule
Week |
Date of Monday |
Topics |
Pages in Hatcher |
Lecture notes |
Example sheet |
Comments |
1 |
Sep. 30 |
Introduction and overview. Topological spaces, bases, products,
subspaces, examples. Homeomorphisms, foundational problem,
invariance of domain, topological invariants. Disjoint union,
quotients, group actions, cut-and-paste. Our friend the square,
the annulus, the Mobius band. Homotopies, gluing lemma.
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1 |
Monday
Tuesday
Thursday
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One
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2 |
Oct. 7 |
Our friend the square. Homotopy equivalence. Paths, "reeling
in", homotopy rel endpoints. Concatenation of paths and
homotopy. Loops, space of loops, concatenation of loops is a
loop, concatenation does not have indentity, inverses, and is
not associative. The fundamental group. Multiplication is
well-defined, closed, has identity, inverses, and
associativity. Basepoints, basepoint isomorphism. Covering
spaces, covering maps, stack of pancakes, covering degree.
Isomorphism of covering spaces. Graphs, the rose with two
petals.
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1 - 4, 21 - 28, 56 - 60 |
Monday
Tuesday
Thursday
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Two
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Note that I have updated the third example sheet - please do
Problem 3.3 and turn it in Friday, noon, in week three.
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3 |
Oct. 14 |
Deck transformations, deck groups. Carpark example. Homotopies
descend, concatenations descend. \(\pi_1(S^1)\) and winding
paths. Definition of \(\Phi\). \(\Phi\) is a homomorphism.
Deck transforms for cover of the circle, straight-line homotopy.
Homotopy lifting and covering spaces. Homotopy lifting property
for covering spaces (1.30). \(\Phi\) is an isomorphism (1.7).
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29 - 31, 60, 71 - 72 |
Monday
Tuesday
Thursday
|
Three
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Fixed typo in Thursday lecture notes - image of lift is in \(Z\). |
4 |
Oct. 21 |
Covering maps have the (unique) homotopy lifting property.
Special case of path lifting. Constant paths lift to constant
paths. Fixed point property. \(B^n\) has the fixed point
property. The tripod space. Induced homomorphisms,
functorality. The fundamental group is a homeomorphism
invariant.
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30 - 31, 60, 34 |
Monday
Tuesday
Thursday
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Four
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5 |
Oct. 28 |
The fundamental group is a homotopy invariant. Retracts,
deformation retracts. No retract theorem. Brouwer fixed point
theorem (for \(n = 2\)). "Tablecloth theorem", Perron-Frobenius
theorem (for \(n = 3\)), fundamental theorem of algebra.
Null-homotopies. Even versus odd functions.
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31 - 34, 36 |
Monday
Tuesday
Thursday
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Five
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6 |
Nov. 4 |
Even and odd functions, Borsuk-Ulam theorem. The fundamental
groups of spheres, of cartesian products. The "Galois
correspondence". Covers induce injections on \(\pi_1\). Index
equals degree. Motivation for free groups. Pointed union.
Beginning free products: words, empty word, length,
concatenation, associativity, reductions.
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31 - 32, 35, 41 - 42, 61 |
Monday
Tuesday
Thursday
|
Six
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7 |
Nov. 11 |
The free products of groups is a group. Earring space.
Statement of Seifert-van Kampen. \( \pi_1(S^1 \vee S^1) = \ZZ
\ast \ZZ\). Factorisations. Diagrams of spaces, diagrams of
groups. Reductions, expansions, exchanges.
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49, 43 - 45 |
Monday
Tuesday
Thursday
|
Seven
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8 |
Nov. 18 |
Finish proof of Seifert-van Kampen. CW complexes: cells,
boundaries, attaching maps, skeleta. Examples. Non-examples:
Cantor set, Earring space, topologist's circle. Properties of
subspaces (precompact, connected, path-connected, Hausdorff,
normal, contractible, simply-connected). Local properties.
Subcomplexes. CW complexes are nice (A1, A3, A4, A5). Spheres
and projective spaces. One-skeleton determines
path-conectivity, two-skeleton determines fundamental group.
Generators and relations, presentations of groups. Finite
generation, finite presentations. Fundamental group of the
rose. Fundamental group of the two-torus.
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45 - 46, 5 - 7, 519 - 523, 39 - 40, 97 (only first paragraph)
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Monday
Tuesday
Thursday
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Eight
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|
9 |
Nov. 25 |
Cell structures on projective plane, sphere, fake rose.
Fundamental groups of CW complexes. Trees, spanning trees.
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49 - 51 |
Monday
Tuesday
Thursday
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Nine
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10 |
Dec. 2 |
Algorithm to compute fundamental group of a CW complex.
Surfaces as CW complexes. Every group is a fundamental group.
Manifolds and CW complexes. Universal covers, examples,
construction, topology, path-connected, simply connected, deck
groups, Galois correspondence.
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49 - 52, 63 - 68, 70 - 72
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Monday
Tuesday
Thursday
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Ten
Eleven
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