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MA40040 diary 2011

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 0: Revision and Examples.  Defn of topological space.
Continuous maps.

Lecture 2: Continuous maps: properties and examples; homeomorphisms,
open maps.  Bases and subbases.  New topologies from old: induced
topology, product topology, quotient topology and their universal
properties. Non-Hausdorff quotient of the reals.

Lecture 3: Examples of quotient topology.  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. Homotopy equivalence and categorical hand-waving to justify
same.

Lecture 8:  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 \Z.

Lecture 10: Map is 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.  S^n simply connected when n>=2.

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
theorem: statement and chat about properties of lifts.

Lecture 13: Path lifting theorem.  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. (Corollary:
\pi_1(RP^n)=Z_2 for n>=2).  Ultimate lifting theorem (statement, idea
and start of proof).

Lecture 15: Ultimate lifting theorem. 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 and a well-defined
map.

Lecture 16: Map is isomorphism.  Universal covering spaces: defn and
uniqueness.  Necessary condition for existence of such.  Chat about
how it is also sufficient.

Lecture 17: Compute \pi_1(RP^n).  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.