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## Filename: diary
## Description: Diary of M40 lectures 2018
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## Modified at: Tue Nov 20 10:59:45 2018
## Modified by: Francis Burstall
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MA40040 diary 2018
Lecture 1: Propaganda: Alg. Toply. is reduction of difficult problems
in topology to problems in Algebra. Example: are S^2 and T^2 homeo?
Strategy for solving this problem. Sample applications:
Fund. Thm. of Algebra; Brouwer Fixed Point Thm, Ham Sandwich Thm.
Chapter 1: Revision and Examples. Defn of topological
space. Continuous maps and homeomorphisms. Open maps.
Lecture 2: Continuous maps: properties and examples. Bases and
subbases. New topologies from old: induced topology,
product topology, quotient topology and their universal
properties. Examples of quotient topology: non-Hausdorff
quotient of the reals; circle as quotient of interval.
Lecture 3: Surfaces as quotients of polygons. Serious
examples: GL(n,R) and other topological groups. Compact open
topology on spaces of continuous maps. RP^n as a quotient
space of R^{n+1}\{0}. RP^n as a quotient space of S^n.
Lecture 4: NEW CHAPTER: Homotopy and the fundamental group.
Connected and path-connected spaces. Locally path-connected spaces.
Based homotopy of paths: defn and proof that this is an
equiv. reln. Product of paths gives well-defined product on
equiv. classes.
Lecture 5: Product of equiv. classes is associative with identities
and inverses where defined. In particular, get a group structure on
the classes of loops with same basepoint: this is the fundamental
group.
Lecture 6: Effect of base-point change on the fundamental group.
Effect of continuous maps. Hand-wave about categories and functors.
Homeomorphic spaces have isomorphic fundamental groups.
Lecture 7: Homotopy of continuous maps; relative homotopy. Homotopic
maps have same induced map on \pi_1 up to isomorphism to fix up base
points.
Lecture 8: Homotopy equivalence. Homotopy equivalent spaces
have isomorphic \pi_1. Examples: balls are homotopy
equivalent to points and so have trivial $\pi_1$.
Deformation retracts. Simply connected and contractible
spaces. The fundamental group of a circle is the integers.
State theorem and draw movies of the proof.
Lecture 9: Lifting lemmata. Define map from \pi_1(S^1) to
the integers.
Lecture 10: Map is an isomorphism. Lebesgue Covering Lemma
(statement). X=U\cup V, with U,V simply connected and U\cap
V non-empty and path-connected, is simply connected.
Application: S^n simply connected when n>=2 (via
stereoprojection).
Lecture 11: \pi_1 of a product is product of
\pi_1 of the factors. \pi_1(T^2)\cong \Z x \Z. Applications:
Fundamental Theorem of Algebra. Brouwer Fixed Point Theorem.
Lecture 12: NEW CHAPTER: Covering spaces, defn and examples
Basic properties of covering maps. Unique lifting
property. Path lifting property (statement and consequences
of uniqueness).
Lecture 13: Path lifting theorem (proof). Homotopy lifting theorem I:
lifting maps of square. Application: Homotopy Lifting II: based
homotopic paths lift to based homotopic paths.
Lecture 14: Applications: covering maps induce injections on \pi_1;
action of \pi_1 on fibres of a covering map. Chat about how this
computes \pi_1(RP^n) for n>=2. Ultimate lifting
theorem (statement and idea).
Lecture 15: Ultimate lifting theorem (proof). Ultimate
Lifting for simply connected domains. Deck translations.
Isomorphism of group of deck translations for a simply
connected and locally path-connected covering space with
\pi_1 of the base: statement; define the map and show it is
a homomorphism.
Lecture 16: Map bijects. Universal covering spaces: defn
and uniqueness. Necessary condition for existence of such.
Chat about how it is also sufficient. Compute \pi_1(RP^n).
Lecture 17: Borsuk-Ulam Theorem and corollaries: a continuous odd fn
S^n\to R^n has a zero, any continuous map S^n\to \R^n has x with
f(x)=f(-x). Ham Sandwich Theorem.
END OF OFFICIAL COURSE!