M8130 Algebraic topology, tutorial 02, 2020
U. 10. 2020
Exercise 1. Show that CPn is a CW-complex.
&y^Uyt faff oic/i^t /f#Zf
(FT 40 d f**A-
.......^rCV.
/» Mil ItoviA
1« -1
•= (p2* " fr~%*r ho-t g2*-1
"T/Jv
2^ /?2a*1
v—v-ftH
	-____                                 - /P?0
	•s.                                                                                                                    ^ f
	^ Jb    \j Jb     i        Ufr  u Sb
	(ft* = ŕ o ju0 =
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
M8130 Algebraic topology, tutorial 02, 2020
U. 10. 2020
Exercise 2. Prove that A := {±,n e N} U {0} as a subspace ofR is not a CW-cornplex. Then show that X := I x {0} U A x I is not a CW-complex either.
«     e1Rj /c £ N*} u io) G™u" ^
C c A <ty    c = ^^u^
0 e A ^ ^ 4 sUt off. oU** ^.4.
/UAP
At ^ A •
	
	(d o) e X0 i    fi^í/^t^w /iviM Mt
	
	LlU              V MO   ÔUotAjiÁ /flfßfoflf.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
M8130 Algebraic topology, tutorial 02, 2020 14. 10. 2020
Exercise 3. Prove that every compact set A in a CW-complex X can have a nonempty intersection with only finitely many cells.
ft mill ¥^ 5*4^ «W/
	
	
	
	
	
	n r. J'ň/* i      'S  «Vi     f/jfcŕd/
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
M8130 Algebraic topology, tutorial 02, 2020
U. 10. 2020
Exercise 4. Show that the Hawaiian earring given by
1 1 X = Ux, y) E IR2, (x--)2 + y2 = —- for some n\
n nz
is not a CW-complex.
(0,0)   /W A   * D-u/t> \ tfM 4wtiȣ ^fito /U   fan. O-uMfi
M8130 Algebraic topology, tutorial 02, 2020 14. 10. 2020
Exercise 5. Show that for a short exact sequence O^A^B^C^Oof abelian groups (or more generally modules over a commutative ring) the following are equivalent:
(1) There exists p : B —>• A such that pf = idA.
(2) There exists q : C —>• B such that gq = idc.
(3) There exist p : B —>• A and q : C —>• B such that fp + qg = idg.
Another equivalent condition is B = A © C', with (p, g) and f + q being the respective inverse isomorphisms.
Prove (1) implies (2) and (3) and (3) implies (1) and (2). The rest is for homework.
0     A A> d fl £ ^°
f     a, M,^ « ^ f
C^irs OA if* ^ —->   ^ —> ^   •--> 2/2
0      A -£> B -•> C ~~>0
= m ftp)
±u)~ f u) °(?-
(d.
ty—
/ e B
5
d-
r
TT
0
Z4-
~ id
k.
	f p* f (j,  - *áB        / %°
	(K l\ß i   (k fyl   - Q/
	
	0          a, Or (L  ~ íAj.oQs   q (o
	i^~<~^                                                 /tum .
	.n.     a b - iáľ
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
M8130 Algebraic topology, tutorial 02, 2020 14. 10. 2020
Exercise 6. Let 0 —> A* A 5* A C* —> 0 be a short exact sequence of chain modules. We have defined the connecting homomorphism <9* : Hn{C) —> Hn-i(A) by the formula <9*[c] = [a], where dc = 0, f(a) = db and g{b) = c. Show that this definition does not depend on a nor b.
0 -> B^i —> 0^ 0
1 ^ I     n        3 i     Qy       I^<k+1 I
1
n^°Vf w    UmC } ~
defalk im. b
|& tW«
7a,:/i  éufc^l*. v>
oS ►
\V I-■y y»-b'
9b, 0b'
	
	--r- -*^-w-^™-^- /i r i   r n     r   ^  - ľfftf"! =0^/4* /M
	9 Lit]   Líj ? L     j ^ l u j         * *
	*                      ^ |.__■—^ . r
	
	1           D/M ^ vi   I í  ^----^ r
	-r? W             ^-C c/J
	C - 0 0 -**"\--^\-Ä—»7-9«—«7-
	
	--^   Ba 1                                         - /9 ŕ//, /Aí
	
	
	^ änn^     jLtw-m,    M Q ŕ, jiaA   n     íkj*a*j\ L   Afr^L     A>Ať ÁRU/
	
	
	
	> M* /M         í#) ^ H* í C)    > ttk-i LA)