Math 527 - Homotopy Theory
Spring 2013
Homework 10 Solutions
Problem 1. Let n 2 [Sorry I forgot to write that on the homework!]. Consider the wedge
X = S 1 S n.
a. Show that the nth homotopy group of X is a free 1 (X )-module on one generator:
n (X ) Z[1
Math 527 - Homotopy Theory
Spring 2013
Homework 6 Solutions
Problem 1. (The Hopf bration) Let S 3 C2 R4 be the unit sphere. Stereographic
=
1
2
projection provides a homeomorphism S = CP , where the North pole corresponds to
[0 : 1] CP 1 . The composite
S
Math 527 - Homotopy Theory
Spring 2013
Homework 5 Solutions
Problem 1. Find a space X such that no choice of basepoint will make it well-pointed.
Solution. Consider the space Q of rational numbers, with its standard metric topology. For
=
any point q Q, t
Math 527 - Homotopy Theory
Spring 2013
Homework 4 Solutions
In Problems 1 and 2, let Top denote the usual category of all topological spaces and continuous
maps between them.
Problem 1. Let U : Top Set denote the underlying set functor.
a. Show that U has
Math 527 - Homotopy Theory
Spring 2013
Homework 2 Solutions
Problem 1. Show that the reduced suspension X = X S 1 of any pointed space X is a
homotopy cogroup object in Top , with structure maps coming from those of S 1 (c.f. Homework
1 Problem 3).
Soluti
Math 527 - Homotopy Theory
Spring 2013
Homework 3 Solutions
Problem 1. An H-space (named after Hopf) is a pointed space (X, e) equipped with a
multiplication map : X X X such that the basepoint e is a two-sided unit up to
pointed homotopy. In other words,
Math 527 - Homotopy Theory
Spring 2013
Homework 7 Solutions
Problem 1. (May 10.7 Problem 2)
a. Let f : X Y be a weak homotopy equivalence. Assuming X is a CW-complex and Y
has the homotopy type of a CW-complex, show that f is a homotopy equivalence.
Solut
Math 527 - Homotopy Theory
Spring 2013
Homework 8 Solutions
Problem 1. Let X be an n-connected space, for some n 0. Show that X admits a CW
approximation with a single 0-cell and cells in dimensions greater than n.
Solution. Since X is n-connected, the in
Math 527 - Homotopy Theory
Spring 2013
Homework 13 Solutions
Denition. A space weakly equivalent to a product of Eilenberg-MacLane spaces is called a
generalized Eilenberg-MacLane space, or GEM for short.
Problem 1. Show that any topological abelian group
Math 527 - Homotopy Theory
Spring 2013
Homework 12 Solutions
Problem 1. Consider the standard inclusions C0 C1 . . . Cn Cn+1 . . . given by
appending a zero in the last coordinate:
z1
z1
z2
z2
.
. . .
.
.
.
zn
zn
0
These give rise to inclusions . .
Math 527 - Homotopy Theory
Spring 2013
Homework 11 Solutions
Problem 1. Show that a path-connected space is weakly equivalent to a product of EilenbergMacLane spaces if and only if it admits a Postnikov tower of principal brations with trivial
k -invarian
Math 527 - Homotopy Theory
Spring 2013
Homework 9 Solutions
Problem 1. Consider the Hopf map : S 3 S 2 .
a. Describe the cober C ( ). It is a familiar space.
Solution. Recall that : S 3
CP 1 was dened as the quotient by the action of S 1 C
(via scalar mul
Math 527 - Homotopy Theory
Spring 2013
Homework 1 Solutions
Problem 1. Show that the following conditions on a topological space X are equivalent.
1. X is contractible.
2. The identity map idX : X X is null-homotopic.
3. For any space Y , every continuous