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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. 1 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. 2 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: 3 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. 3