ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENT ATI ONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATE MAT YCZNE XIX (1976)
W. K
u l p a(Katowice)
Another proof of a Hurewicz theorem
An elementary proof of the Hurewicz Theorem on isomorphism is given; if a space X is (n — 1)-connected, then л п(Х) яа Я я[Х), 2.
Let I
9be the q-th dimensional cube, 1° = {0}, I q(n) be the w-th dimensional skeleton of I
9and let I
9= I 9(q —
1). Define е%в: I
9~>I9+i by (h, . . . , t q)h+{tlf ..., ti_1} e, t{, . . . , t q),
8= 0,1, and define = ao e j8, where a : I 9+1->X is a map. Let Qq{X), Qq{X, x0, n), Qq{X) mean free abelian groups generated, respectively; (a) by the all continuous maps a:
I
9-^X, (b) by the all continuous maps a: I
9, 1
9(n) ->X, x0, where x Q is a fixed point of the space X , (c) by continuous maps a: I
9-»X such th a t o{tx, ..., tq) = cr(0, t2, . . . , t q) for each txe [0 ,1 ].
P u t Cq(X) = Q q(X)/Qq(X), q > 1, CQ(X) = Q
0(X) and Cq( X , x
0,n) Qq(X^ /Qq( X )X
q, n)r}Qq(X)j q ^ 1, C
q( X , xq,
ti) = Qq{X, x qj fi).
Define boundary homomorphisins dq: Gq{X)->Cq_x{X) and dq: Ca{X, x Q, n) -> Cq_x{X, x 0, n) by the formula
i —1
where a: I q ->X is a generator. Let H q(X) — Hg[C(X)) and H q{X \n )
= H q(C(X, x0, n)) be the g-th homology groups. The functor H-C of the cube homology theory is equivalent to the singular homology theory.
In this paper a notion of the q-th homotopy group nq{X) — nq{X, x0) is such as in [2] with a multiplication induced by *:
[ a(
2t
1 7 12, .. ., tq) i f 2 t 1< l , (a*rj) (tx, ..., ta) =
* \ v P h - l
If q ^ 2 , then nq{X) is an abelian group (see [2], IY 3.1).
In a following lemma we shall use an easy to show fact th a t if nq{X)
= 0, then each continuous map a: I
9+1->X has an extension a : I
9+1->X
(see [1], I I 1.12).
86 W. K u l p a
L
emma1 (see [1], § 8.8). I f a space is n-connected, then the embedding iq: Cq(X, x 0, n) <= Cq(X) induces an isomorphism iq*: H q(X, n) ^ H q(X) for each q > 0.
P ro o f. I t suffices to find functions Dq which assign to each contin
uous map a: I q ->X a continuous map Dq{a): I
q+1-+-X satisfying following conditions :
(1) [-»„(<r)]S+1>0 = a, (2) ^ , ( 4 - 1 ) =
(3) if otQ',(X), then Dt (o)'Q't (X), (4) Ta(o)€Qt ( X , æ„, n),
(5) if aeQq(X, æ0, n), then rq(a) — a, where rq(a) = [Bq(a)]
Since, according to (2)
= [[•0„(ff)]J+1,,]i,-i = [ [ А г И Г ] ^ =
« +1,1
a
hence we obtain dr = rd. This, and (3) implies th at, r is a chain homo
morphism. From (1), (2) and (4) it follows th a t dq+
1Dq{a)
= У ( - [ A + ) ] J + ) + ( - i ) a + ‘ ( [ Д Л ^ Г 1'0 - C-Da (< r)3 5 + 1 '1)
ï—1 a
i—l
= D ^ r J a ) + ( -1 )« +1 [lJa) - iqrq(a)].
How, according to (5) we obtain dD — Dd = ( — l ) e (1 — гг) and ri = 1.
Hence, iqm: H q(X, n) - ^ H q(X) is an isomorphism.
How, we shall prove an existence of such Dq, q — 0,1, ... satisfying conditions 1-5.
(a) If а: I й ->X is such th at aeQq(X, a?0, n), then we put Dq{a) = O’p, where p: I
q+1->Iq is the projection.
How, let us assume th a t o
4Qq(X, x0, n).
(b) Let a: 1° -+X, aaX. Since X is path-connected, hence there exists a path D
0{o): I such th a t D
0(a) (0) = a and B
0(a) (1) = x 0.
(c) Let q < n. Let us assume th a t for each p < q, Dp is defined such th a t it satisfies conditions 1-4 and let
04Qq(X), o: I q ~^X. Conditions (1), (2) and (4) uniquely determine Da(a)\jq+i. We put Dq(a) to be an extension of Dq(o)\jq+i. Such an extension exists because nq(X) = 0.
(d) How let us assume th a t q > n and th a t Dp is defined for each
p < q. Let
04Qq(X), a: I q ->X. B q(o)\iqx{0]^jqxl is uniquely determined
by conditions (1) and (2). Since q > n hence I q+1(n) c I q x {0} и I® x I .
Proof of Hurewioz theorem 87
So, it suffices to pu t Dq(o) = [Bq(a)\iaX{o}^i<ixi)'ri where r: I
9+1I й x x { 0 } u la x l is a retaction.
(e) Let о-c Qq(X). Let us assume th a t for each p < q_, B p is defined such th a t it satisfies conditions 1 -4 . В а(а
)\1ах{0^ ^ х 1is uniquely deter - mined by conditions (1) and (2). P u t B q{ a) (tx, ..., tq+l) = (De(or)|jS><{0}^ a><J)-
•(0,t2, . . . , t q+1). According to (2) and (4) it is easy to see th a t Dq(e) is well-defined.
Let Qn{X) be the %-loop space, i.e. Qn{X) = {I n, i n -^X, oo0}, with compact-open topology. Define a homomorphism <pn: nn{X) л х[й
п~ 1(X)),
2, by <pn{o) (t) (tx, ..., tn_x) = a{t, tx, . . . , t n_x). I t is easy to see th a t <pn is an isomorphism. Analogously, define an isomorphism yn: H n(X, n — 1) - * H x(Qn- l {X),^ 0) induced by y>n{a) (t) (tx, ..., tn_x) = a(t, tx, . . . , t n_x), where a: I n, i n->X, x 0.
The Hurewicz homomorphism hn: л п(Х)-^-Нп(Х) is a composition я . ( Х ) - ^ Н п( Х , п - 1 ) ^ Е л(Х), W ^ [ . ] »
hW , [ « V 2 . ( I , n - X ) , [<r]
L
emma2 (see [1], 8.8.3). I f a space X is path-connected, then the homo
morphism hx: л х(Х) ~^Hx(X, 0) is an epimorphism and k e r ^ = \лх{Х),
H
ixbewiczT
heorem. I f a space X is (n — 1)- connected, n > 2, then hn:
л п(Х) ->Hn{X) is an isomorphism.
P ro o f. Let us consider a diagram жп( Х ) --- >----
I*]
The diagram is commutative. To see th a t hn is a homomorphism, it suffices to verify th a t hn is a homomorphism. But hn{a*r}} — Vn'htdcpnio)} {(pn(r})}) = Vn'dV nWo + bPnWlo) = Mo + h]o = K { o } + h n {rj}. If n ^ 2, then л п(Х) is an abelian group, and hence л х[йп~х(Х)) is an abelian. So, by Lemma 2, hx is an isomorphism. This implies th a t hn is an isomorphism. Now, to show th a t hn is an isomorphism it suffices to verify th a t in* is an isomorphism. But this follows immediately from Lemma 1.
The author wishes to express his gratitude to Professor J. Dugundji for his help during preparation of this note.
SILESIAN UNIVERSITY, KATOWICE
References