Algebraic Topology
Spring 2009
International Master Class
Institute of Mathematics
Vietnam Academy of Science and Technology
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TIME and PLACE
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You can find the time-schedule for the
course at the Center for Postgarduate Training, Inst. of
Math. The first lecture will be held
on Wednesday, January 14, 2009.
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Lecture hall 5, Building A14
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INSTRUCTOR
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Prof. Nguyen Viet Dung, Institute of
Mathematics, VAST
Prof. Lionel Schwartz, Univertsité Paris 13
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Contents of the course
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Chapter 0: Generalities
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1.0
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Generalities on categories,
functors and natural transformations. The Yoneda lemma.
Examples.
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2.0
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Recollections on general
topology : quotient topology, filtration (direct limit
topology), the compact open topology on mapping spaces.
Connected spaces and arcwise-connected (or
path-connected) spaces. Definition of $\pi_0(X)$.
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3.0
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Actions of topological
groups : essentially proper free actions of locally
compact groups on locally compact spaces.
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4.0
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Examples of spaces : the
spheres, the classical groups, the grassmanians, (in
particular projective spaces) $S^\infty, RP^\infty, CP^\infty$.
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5.0
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Graphs, the Euler
characteristic of a graph.
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6.0
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Cell complexes (definition and
examples).
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Chapter 1: The Fundamental Group
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1.1
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Definition of based and unbased
homtopy. Definition of contractible space (in particular
$S^\infty$ is contractible). Definition of a retract, of a
deformation retract.
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1.2
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Definition of the fundamental group
$\pi_1(X,x_0)$ as a functor on pointed spaces. Examples
: the fundamental of an H-space is abelian.
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1.3
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Computation of $\pi_1(S^1)$,
Applications : Brouwer, d'Alembert...
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1.4
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The set of free homotopy classes from
$S^1$ into a space as the set of conjugacy classes in
the fundamental group.
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1.5
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The fundamental groupoid.
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1.6
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Definition of higher homotopy groups,
examples using the loop space. $\pi_k(S^n)$, $k<n$.
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Chapter 2: The Van Kampen
Theorem
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2.1
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The Van Kampen theorem.
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2.2
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Proof of the Van Kampen theorem.
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2.3
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Applications : the Nielsen-Schreier
theorem. Various computations.
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Chapter 3: Covering Spaces
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3.1
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Definitions and examples.
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3.2
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Definition of a fibration, a covering
space is a fibration.
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3.3
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The (unique) homotopy lifting property
for covering spaces.
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3.4
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Morphisms of covering spaces.
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Chapter 4: Covering Spaces:
classification
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4.1
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Galois covering spaces.
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4.2
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The equivalence of categories between
connected covering spaces over $B$ (conneceted) and
$\pi_1(B)$-sets.
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4.3
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Applications.
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4.4
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The Van Kampen theorem again.
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Chapter 5: Cofibrations
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5.1
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Definition of cofibrations and of the
HEP (homotopy extension property)
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5.2
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CW-complexes, a CW-pair is a
cofibration, applications to quotient spaces.
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5.3
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Attaching cell to spaces, the behaviour of homotopy groups. Construction of $K(G,1)$.
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5.4
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Relative homotopy groups.
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5.5
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The compression theorem and Whitehead
theorem.
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If we have enough time we want to do the
following
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Chapter 6: Additional Topics
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6.1
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The Blackers-Massey theorem
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6.2
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Computation of $\pi_n(S^n)$
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6.3
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The nerve of a category and $K(G,1)$.
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References
The main textbook will be
C. Godbillon Eléménts de topologie
algébrique, Hermann 1998 (in French).
Otherwise we will use
A. Hatcher, Algebraic Topology (available
at http://www.math.cornell.edu/~hatcher/AT/AT.pdf)
E. Spanier, Algebraic Topology, Springer
1994
M. Zisman, Topologie algébriqie élémentaire,
Armand Colin 1972 (in French)
Exercises
sheet 1
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