Download Higher Homotopy Groups

Higher Homotopy Groups
Ye Shengkui
National University of Singapore
October 16, 2009
Abstract
In this note, several fundamental facts about higher homotopy
groups are presented.
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Motivation
A fundamental problem in algebraic topology is to determine when a map
between two spaces f : X ! Y is a homotopic equivalence. The Whitehead
theorem (see Section 3) answers this problem completely when X and Y are
simplicial complex (more generally, CW complex) using higher homotopy
groups. To avoid troubles, all spaces in this note are assumed to be "good"
spaces (for example, simplicial complex, CW complex ) except for Section
2, where the de…nition can be given for a general space.
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What Are Homotopy Groups?
De…nition 1 Let X be a topological space with base point x0 ; n be a positive
integer and I n = I I ::: I for I = [0; 1]: n (X; x0 ) = [(I n ; #I n ); (X; x0 )]
is the set of homotopy class of maps from (I n ; #I n ) to (X; x0 ): For any
f; g 2 n (X; x0 ); de…nte f g : (I n ; #I n ) ! (X; x0 ) by
f (t1 ;
(f
g)(t1 ;
; 2tn ); 0
; tn ) =
g(t1 ;
; 2tn
1);
1
;
2
tn
1
2
tn
1:
This is well-de…ned. The set n (X; x0 ) is a group under this multiplicity ;
called the n-th homotopy group of (X; x0 ).
1
Proposition 2 When n = 1; n (X; x0 ) is the fundamental group of the
space (X; x0 ): When n 2; n (X; x0 ) is an abelian groups.
Proof. The …rst statement is obvious from the de…nition. When n
f; g be two elements in n (X; x0 ). Then
2; let
:
2.1
Why need Higher homotopy Groups?
Theorem 3 (Whitehead theorem) Let f : X ! Y be a map between connected simplicial complex. Then f is homotopy equivalence i¤ fn : n (X; x0 ) !
n (Y; y0 ) are isomorphism for all n:
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How to use Homtopy Groups?
De…nition 4 (Fibration) Let p : E ! B be a map of two path connected
spaces. If for any space W and maps f : W ! E and g : W I ! B such
that p f = g inl; where inl is the inclusion: W = W f0g ,! W I; there
exist a map g~ : W I ! E such that the following diagram is commutive
f
W
!
E
#
% g~ # p
g
W I !
B
then p is called a fribration and F = p
called the …ber of the …bration.
1 (b
0 );
the preimagie of base point, is
Proposition 5 Covering spaces E ! B and …ber bundles are …brations.
Theorem 6 If F ! E ! B is a …bration, then the following sequence is
exact
!
n+1 (B)
!
n (F )
!
n (E)
!
n (B)
2
!
n 1 (F )
!
!
0F
!
0E
! 0:
Corollary 7 For covering space p : E ! B; pn : n (E) ! n (B) is isomorphism for all n
2: In particular, n (Sg ) = 0 ( n
2) for a closed
surface of positive genus g:
Proof. The …ber F of p is a discrete space, which implies all the higher
homotopy groups of F are trivial. From the exact sequence above, the
statement is proved. From the exact sequence, we can see also that 1 (E)
is a subgroup of 1 (B): For any closed surface Sg of positive genus, there is
universal covering R2 ! Sg : Since R2 is contractible, all the higher homotopy
groups of Sg are vanishing.
Any element f : (I n ; #I n ) ! (X; x0 ) in n (X; x0 ) induces a map f :
(S n ; s0 ) ! (X; x0 ); which can be viewed as an element in Hn (X; x0 ): It can
be checked that this de…nes a map (called Hurewicz map) from n (X; x0 ) to
Hn (X; x0 ): The following proposition relates the homotopy groups and the
homology groups.
Proposition 8 Let X be a connected space with i (X) = 0 for 0 < i < n;
i.e. X is n 1 connected, then the Hurewicz map induces an isomorphism:
n (X) ! Hn (X):
i (S
Corollary 9
n)
= 0 when i < n;
n (S
n)
= Z:
i (S
1)
= 0 when i > 1:
Proof. The …rst statement can be proved using the above proposition once
and once again. Note that there is universal covering R1 ! S 1 : By Corollary
7, the second statement is proved.
In general, the higher homotopy groups of the sphere S n ( n 2 ) are
very complicated. Actually there are all …nite in su¢ cient large dimension and nonvanishing in in…nite dimensions. The following example shows
2
3 (S ) is not trivial.
Example 10
3 (S
2)
= Z:
Proof. Let S 3 ! S 2 be the Hopf bundle with …ber S 1 . By the exact
sequence in Theorem 6 and Corolarry 9, we have
3 (S
1
which shows that
)=0!
3 (S
2)
3 (S
3
)=Z!
3 (S
2
)!
2 (S
1
) = 0;
= Z:
All the results and their proofs can be found in the book of "Algebraic
topology" by Allen Hatcher.
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