Math 753: Week 1
Laureniu Maxim
October 5, 2013
Chapter 1
Basics of Homotopy Theory
1.1
Homotopy Groups
Denition 1.1.1. For each n 0 and X a topological space with x0 X , the n-th homotopy
group of X is dened as
n (X, x0 ) = f : (I n , I n ) (X, x0 ) /
w
Algebraic Topology III Notes
10/15/2013
1
Leray-Serre Spectral Sequence
Theorem 1. Let F i / E / B be a bration with 1 (B ) = 0 and 0 (E ) = 0. Then,
there is a spectral sequence with E 2 -term
2
Ep,q = Hp (B, Hq (F )
which converges to H (E ).
So, there
1
Finishing Up CW Approximations
Lemma 1.1. Let (X, A) (X , A ) be a map of pairs, where A, A are CW complexes. Letet
(Z, A) be an n-connected model of (X, A) with associated map f : (Z, A) (X, A), and, let
(Z , A ) be an n -connected model of (X , A ) wi
1
Whiteheads theorem.
Statement: If f : X Y is a map of CW complexes inducing isomorphisms on all homotopy
groups, then f is a homotopy equivalence. Moreover, if f is the inclusion of a subcomplex X
in Y , then there is a deformation retract of Y onto X .
MATH 753 WEEK 2
NOTES BY MEGAN MAGUIRE
1. Relative homotopy groups
Let x0 A X, and n Z with n 1. Let
I n1 = cfw_(s1 , . . . , sn ) I n : sn = 0.
Dene J n1 by
J n1 := I n I n1 .
Then we dene the nth homotopy group of the pair (X, A) with basepoint x0 to be
MATH 753 NOTES OCT 17
Example:
We will compute H (S n ) for n > 1. If we look at the spectral sequence for the path bration
S n P S n S n
=
Hq (S n )
, p = 0, n
0
, otherwise
2
Ep,q = Hp (S n ; Hq (S n ) =
will converge to H (P S n ) = H (point)
H (S n )
Cohomology groups of Lens spaces
Consider the scaling action of C on Cn+1 \cfw_0 S 2n+1 , n 1. By identifying Z/q with the q th roots of
unity in C we get an action of Z/q on S 2n+1 . We call the quotient L(n, q ) a Lens Space. We allow n = .
The action o
INTRODUCTORY TOPOLOGY
MATH 753
PROBLEM SET #1
1. Use homotopy groups in order to show that there is no retraction RPn RPk if
n > k > 0.
2. Show that an n-connected, n-dimensional CW complex is contractible.
3. (Extension Lemma)
Given a CW pair (X, A) and
INTRODUCTORY TOPOLOGY
MATH 753
PROBLEM SET #2
1. Show that if S m S n S l is a bration, then n = m + l and l = m + 1.
2. Compute the cohomology groups of SO(4).
3. Compute the cohomology of the space of continuous maps S 1 S 3 .
4. Find the ring structure
INTRODUCTORY TOPOLOGY
MATH 753
PROBLEM SET #3
1. Let : E B be a principal S 1 -bundle, with 1 (B ) = 0. Consider the cohomology Serre spectral sequence associated to , and let a H 1 (S 1 ) be a generator.
Show that the rst Chern class of can be computed b