M8130 Algebraic topology, tutorial 08, 2020
2. 12. 2020
Define the n-th homotopy group of the space X with the base point x0 as the group of homotopy classes of maps (In,dln) ->■ (X,x0) with the operation given by prescription:
t r> }
f(2tuh,...,tn) 0<U<±
^(2ti-l,t2,...,tn) \<ti<\.
Denote it nn(X,x0).
Exercise 1. Show the operation on -nn(X,x0) is associative.
^
I
r (t<)
i
^ f fa) eJT
M8130 Algebraic topology, tutorial 08, 2020_2. 12. 2020
Exercise 2. Show that the element given by prescription
is really the inverse element of f.
M8130 Algebraic topology, tutorial 08, 2020
2. 12. 2020
There is a long exact sequence:
•••->• 7ln+1(X,A,X0) 7ln(A,X0) 7ln(X,X0) 7ln(X,A,X0) 7tn_i(A,X0) ->-...
Exercise 3. 57jou; £/«e exactness of this sequence in nn(X, A, x0) and nn(A, x0).
9
~ 0
7
I*
n	— 1		/I    r* r
			
i- A			
	---- 4r		
	—>		
			
			
	j-,		
			
			
			
			
			
			
	é—       / '----	—>^	
			
			
			
			
			
	J  i _ru ja _l               n i /í-             ^              i /             X                        *                 _m                m   mŕk ""1		
	NOW    /v^- Aw** /O    j(IXj0W£> AAfrtP		
			
	/)         ^        í/ň a r	/?   ,    ff«   3 7^ -} //V   y l	
		f • ^ '     '      ( ' v	
			
	j            V. "      (             /                   ' ' /		
	C' o -f A/ Č0p(*6		
	1		
	Ji/   * ■ ť". $r: r \		
		//NT .	
	A    rJjyiöjyr?    /.i      TT   / VL VC-)       ^    £ ^^flrf'		
			
M8130 Algebraic topology, tutorial 08, 2020
2. 12. 2020
Exercise 4. Show that every fibre bundle is a fibration.
, •> -A
H(*,t) «* [<*(*(:), t^)
As
	T	/ (	
	_L	Í   ^1_i	---- A&e,   <£>čl/ á/ps-s-s
			-- -■ r i--
		~LĹJJJJJi<	
	±		
			
			
			r--o,    í,       , s r.
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
			
M8130 Algebraic topology, tutorial 08, 2020
2. 12. 2020
Exercise 5. Show the structure of the fibre bundle Sr'
lPn.
its
i
flu) 2 ft* £°
(4>S° ->f(u) * £
M8130 Algebraic topology, tutorial 08, 2020 2. 12. 2020
Exercise 6. Show the structure of the fibre bundle S2n+1       CPn with the fibre S1.