NOTES ON THE COURSE ALGEBRAIC TOPOLOGY
BORIS BOTVINNIK, EDITED BY HAL SADOFSKY
Contents
1. Important examples of topological spaces
1.1. Euclidian space, spheres, disks.
1.2. Real projective spaces.
1.3. Complex projective spaces.
1.4. Grassmann manifolds
Algebraic Topology (634), Final exam, December 6th.
(1) Calculate H (CP n ).
SOLUTION: Induct on n. CP 1 S 2 . So Hi (CP 1 ) is Z for i = 0, 2 and
=
0 else. Notice the Zs occur just where there are cells in the CW structure.
Inductively assume (n > 1)
Hi
Algebraic Topology (634), Solutions to Homework 1.
3.4 Let Id denote the identity map of X . Suppose X is contractible. Then
Id cx0 for some x0 X . Dene i : X by i() = x0 , and j : X
as the only map it can be.
Then j i = Id . So we just have to check tha
Algebraic Topology (634), Homework 2, solutions.
(1) Prove that if f g (as maps) then f h g h, and that k f k g .
SOLUTION: Let X and Y be the domain and range of f, g . There is a
homotopy H : X I Y so that H (x, 0) = f (x), H (x, 1) = g (x).
Let h : Z X
Algebraic Topology (634), Homework 3, due October 13 in class.
(1) Give a CW structure for CP n .
SOLUTION:By induction. CP 0 = . Suppose CP n1 is a CW complex
with one cell in each dimension 2j for 0 j n 1. Let
f : S 2n1 CP n1
be dened by f (z1 , . . . ,
Algebraic Topology (634), Homework 4, due October 20 in class.
(1) Prove that if we have f : X Y , g, h : Y X then if f g 1Y and
hf 1X then f is a homotopy equivalence from X to Y .
SOLUTION: Note
g = 1X g (hf )g = h(f g ) h 1Y = h
So we get f g 1Y by hyp
Algebraic Topology (634), Homework 5, due October 27 in class.
For week 5, read chapter 6, keep in mind there is a midterm on Friday. It will cover
homework through this homework. Homotopy, homotopy equivalence, retracts,
strong deformation retracts, proj
Algebraic Topology (634), Homework 6, due November 5 in class.
(1) Let G = (E, V ) be a graph (here E is the set of edges, and V is the set of vertices, and there
is a function v : E V V taking an edge to a pair of vertices). Describe a CW complex
X (G) t
Algebraic Topology (634), Homework 7, due November 10 in class.
(1) Let (n, k ) have the values (4, 2), (5, 3), (6, 2), (6, 3). For each of these values
of (n, k ) give a table showing the number of cells of each dimension. Can
you see any simply describe
Algebraic Topology (634), Homework 8, due November 17 in class.
(1) The standard n-simplex, n Rn+1 , is the convex hull of the vertices
cfw_e1 = (1, 0, 0, . . . ), e2 = (0, 1, 0, 0, . . . ), . . . en+1 = (0, . . . , 0, 1).
A non-degenerate ane n-simplex i
Algebraic Topology (634), Homework 9, due November 24 in class.
(1) A covering p : Y X is regular if p 1 (Y, y0 ) is a normal subgroup of
1 (X, x0 ). Prove that p is regular i given any y1 p1 (x0 ), there is a
homeomorphism f : Y Y so that pf = p and f (y
Algebraic Topology (634), Homework 10, due December 1 in class.
-Complexes
A -Complex is in between a simplicial complex and a CW complex. It has enough
structure so that you can calculate simplicial homology, but it is not required that
each simplex be d
Final Review, 634
(1) Is 1 (RP 2 RP 2 ) nite or innite? If nite, prove that. If innite, nd
an element of innite order.
(2) Know the fundamental group of a CW complex in terms of generators and
relations.
(3) Describe the universal covering space of a path