An analogue of KÜnneth theorem in strong homology

2002,22B(1):32-40

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AN ANALOGUE OF KUNNETH THEOREM IN STRONG HOMOLOGY 1 Dae- Woong Lee Department of Mathematics, Myungshin University, Sunchon, Chonnam 5,40-871, South Korea E-mail: dwleellmath.chonbuk.ac.kr

Abstract After defining the strong tensor product of strong (sub)chain complexes, it is shown that an analogue of the Kiinneth theorem holds in strong homology by proving that the kernel (cokernel) of connecting homomorphisms is isomorphic to the direct sum of torsion (tensor) products of strong homology groups. An isomorphism between strong (r-stage) homology groups of inverse systems is also constructed. Key words Derived limit, strong (r-stage) homology, strong tensor product, MittagLeffler property 1991 MR Subject Classification

1

55U25, 55N35, 55P55

Introduction Let G and Z be an abelian group and the set of all integers respectively.

And let

C = (C-y, In" f) denote an inverse system of chain complexes C; and chain maps l-y-y' : C-y' -> C-y,"I ::; "I' indexed by a directed set f. Strong homology groups Hp(C) of the inverse system C were defined by J. T. Lisica and S. Mardesic(7). Using the ANR-resolution[9,1l], algebraic topologists defined the strong homology group Hp(X; G) of a topological space X with coefficients in G and gave examples to compute the strong homology group of topological spaces. S. Mardesic and T. Watanabe[12] proved that the strong homology group Hp(X; G) of a space X vanishes if p is greater than the shape dimension sdX of X and showed that Hp (X; G) coincides with the tech homology group Hp(X; G) if p is equal to the shape dimension of X. Using the continuum hypothesis, S. Mardesic and A. V. Prasolov[9] showed that the group Hp(X; G) does not satisfy the Milnor's additivity axiom and does not have the compact support axiom. C. A. McGibbon and R. Steiner proposed some questions[13,14] about the first derived limit of the inverse limit to control the phantom map easily. The author has found some properties of the strong homology groups and derived limits of inverse systems[3-6]. In 1998 the structure theorem and universal coefficient formula with respect to the strong homology group Hp(X; G) and the tech cohomology group jfP(X;G) were constructed by S. Mardesic and A. V. Prasolov[lO]. The purpose of this paper is to show that if C = (C-y,

In" f)

is an inverse system of free

abelian chain complexes and if (C* (C'), d') is any strong chain complex, then there exists a 1 Received

December 6,1999; revised February 10,2001. The author was supported by the KOSEF, 1999

33

Lee: ANALOGUE OF KUNNETH THEOREM IN STRONG HOMOLOGY

No.1

short exact sequence 0-> coker(c n) -> Hn(fJ.(C) ® t.(C')) -> ker(Cn _1) -> in which ker(Cn _1) ==

EB

°

Tor(fI;(C), fIq(C'))

p+q=n-l

and coker(c n) ==

EB

p+q=n

fI;(C) ® fIq(C'),

where C n : H n+ 1(B- ® t.(C')) -> Hn(K ® t.(C')) is a connecting homomorphism induced by a short exact sequence of chain complexes. We also show that if C = (CI ' f l l " I'] is an inverse system of chain complexes with dimension dim C = m 2:: 0, then the homomorphism

induced by a projection is an isomorphism for r > m - p. Throughout this paper, II and EB mean the direct product and the direct sum respectively.

=

And for each 1 ('Yo, 'Y1l ... , 'Yn) E I'", we associate an object at by the object 010 of the first index 'Yo in some categories (e.g. the category of groups or chain complexes), i.e., at = 010'

2

The Construction of Exact Sequences Let 9

= (GI , gil', I') be an inverse system of abelian groups G I

°

and group homomorphisms

gil' : GI , -> GI , 'Y :S 'Y' over the directed set r. And let I'", n 2:: be the set of all increasing sequences 1 = ('}'O,'Yl,"','Yn),'YO:S 'Yl :S ... :S 'Yn,'Yi E r and let 1j Ern-I, 0:S j:S n be

obtained from 1 E I'" by deleting the j-th factor 'Yj, i.e., 1j We define n-cochain groups Cn(Q) of 9 by

Cn(Q) =

= ('Yo, 'Yl,... ,'Yj-1l 'Yj+ll"

. ,'Yn).

II G'f' n 2:: 0,

tErn

where G t is the abelian group G-io as noted earlier. Let prt : C"(Q) -> G t be a projection. The coboundary operators bn

cn-l(Q)->

cn(Q), n 2:: 1, are defined by

prt(b"y) where Y E cn-l(Q). For n

"

= gIOll(prtO(Y)) + ~)-l)jprtj(Y)'

= 0, we put bO =

The n-th derived limit limn

g, n 2::

°:°

j=1

-> C°(Q). Then we have a cochain complex

°([9],[12],[15]) of 9 is defined by

We can see that limo 9 is equal to the inverse limit limg of the inverse system g.

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ACTA MATHEMATIC A SCIENTIA

The p-th strong chain group complexes is given by

Cp ( C), p E Z, ofthe inverse system C = (Con In" I') of chain 00

Cp(C)

= II

II (Cy)p+n, t = (-ro,'" ,'Yn),

n=O'j'EI'n

where (C'j' )p+n is a (p + n )-chain group of the chain complex Cia' dp : Cp+1(C) -+ Cp(C) is defined by

pr(;o)(dp(x))

Vo1.22 Ser.B

= o(pr(;o)(x))

for n

A boundary operator

= 0,

{ (-l)npr'j'(dpx) = o(pr'j'(x)) - liOi'(pr'j'O(x)) - 2:;=1 (-l)jpr'j'j (x) where x E Cp+1(C) and pr'j' : Cp(C) complex; -

(C.(C),d): ...

-+

for n ~ 1,

(Cio)p+n is a projection. Then we have a strong chain

d d,-+ Cp+1(C)~Cp(C)..!=.Cp_1(C) -+ ....

The p-th strong homology group Hp(C),p E Z, of the inverse system C is defined by

We define a strong subchain complex (D.(C), d) of (C.(C),d) by a chain complex whose p-th strong chain group is a subgroup

Dp(C) =

EB EB (C'j' )p+n, t = ('Yo,"', 'Yn) 00

n=O'j'EI'n

of Cp(C), and its boundary operator is the restriction of dp,p E Z. We denote the group H;(C) as the homology group Hp«D.(C),d)) of the strong subchain complex (D.(C), d). We also define the p-th relative strong chain group Cp ( C, s), P E Z, as the following:

We then obtain a relative chain complex (Cu(C, s), d), where d is a boundary operator induced by d. And we can define: Definition 2.1 The p-th relative strong homology group Hp(C, s),p E Z, is defined by

Proposition 2.2 Let C = (Ci we have a long exact sequence

,

In'' I') be an inverse system of chain complexes. Then

of strong homology groups. Proof Since the sequence

is exact for each p in Z. we have the long exact sequence of strong homology groups just like the ordinary homology groups[1,161.

Lee: ANALOGUE OF KUNNETH THEOREM IN STRONG HOMOLOGY

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35

Let (C*(C'), d') be any strong chain complex. The strong tensor product

Definition 2.3

(D*(C), d)® (C*(C'),d') of (D*(C), d) and (C*(C'),d') is defined by the chain complex (D*(C) ® C*(C'),d) whose n-th chain group is defined by (D*(C) ® C*(C'))n = and its boundary operator

d is

E9

p+q=n

Dp(C) ® Cq(C')

defined by

where cp E Dp(C) and c~ E Cq(C'). Let Hn(D*(C) ® C*(C')) denote the homology of this chain complex (D*(C) e C*(C'), d). In the strong subchain complex (D*(C),d) of(C*(C),d), we denote (K,O) and (B-,O) as the chain complexes of cycles and boundaries whose p-th chain groups are K p and B; = B p - 1 respectively, where the boundary operators Lemma 2.4

Let C = (Ci

,

fn', I')

°are the trivial homomorphisms.

be an inverse system of free abelian chain complexes.

If (D*(C),d) is a strong subchain complex of (C*(C),d) and if (C*(C'),d') is any strong chain complex, then there exists a long exact sequence ... -+

E9

p+q=n+1 -+

B;

E9

®

p+q=n

Hq(C')

-+

E9

p+q=n

B; ® Hq(C')

-+

s, ® Hq(C') -+ Hn(D*(C) ® C*(C'))

E9

p+q=n-1

K p ® Hq(C') -+

...

of strong tensor products, where p and q are integers.

Proof Since the direct sum of a family of free abelian groups is also free abelian, and every subgroup of a free abelian group is itself free[2], we obtain a split exact sequence

of free chain complexes. By tensoring with (C*(C'),d'), we have the following exact sequence of chain complexes

0-+ (K,O) e (C*(C'),d') ~(D*(C),d) ® (C*(C'),d')

~(B-,O) e (C,,(C'),d')

-+

O.

For each i E I'" and i' E r,n, the boundary operators of the second and the fourth chain complexes have the following forms (pr't ® prt' )(d(kp ® c~)) = (-l)Pprt(k p ) e prt/(d'(c~)); { (pri ® prt' )(d(b e c~)) = (-l)Ppri(b ) e prt/(d'(c~)), p p respectively, where kp E K p, bp E B; and c~ E Cq(C'). From the short exact sequence of chain complexes, we obtain a long exact homology sequence

(A)

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ACTA MATHEMATIC A SCIENTIA

Vo1.22 Ser.B

where Cn and Cn-l are connecting homomorphisms. We now show that there exist two isomorphisms:

EB B; ® Hq(C') ~ Hn(B- ® C.(C')); { EB s, ® Hq(C') ~ Hn(K ® C.(C')). p+q=n

p+q=n

Similarly, let K~ and B~ be the groups of strong q-cycles and strong' q-boundaries of (C.(C'), d') respectively. We consider the following horizontal and the second vertical exact sequences:

0

B; ® Cq+l (C')

! 10i'

B-®K' p q

! 100'

B; ® Cq(C')

--+

B; ® B~_1

--+

! 1 ®d'

--+

0,

! 1 ® iB

B; ® Cq- 1(C')

B;

e Cq- 1(C')

where i' and i B are inclusions. Since 1 ® i' and 1 ® iB are monomorphisms, we have ker(l

e d') ~ ker(l ® s') ~ B; ® K~

and im(l ® d') ~ B;

e B~_I.

Hence the groups of n-cycles and n-boundaries of the chain complex (B-, 0) ® (C.(C'), d') are

respectively. Thus Hn(B- ® C.(C')) =

EB

p+q=n

B; ® K~ /

EB

p+q=n

B; ® B~.

Let us now consider a short exact sequence

where j' is an inclusion and t' is a projection. Since

B;

is free, we also have a short exact

sequence

0-

EB

p+q=n

B;

e B~ ~

EB

p+q=n

B; ® K~

~

EB

p+q=n

B; ® Hq(C') -

Thus we have

EB

p+q=n

B; ® Hq(C') ~ coker(l ® j') . ffi ® K'q / ffi ® B'q \J7 Bp \J7 Bp p+q=n p+q=n = Hn(B- e C.(C')), ~ -

o.

Lee: ANALOGUE OF KUNNETH THEOREM IN STRONG HOMOLOGY

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37

for each n E Z. Similarly, we also have

EB s; 0 Hq(C') == H,,(K 0 C*(C')),

p+q="

for each n E Z. From these facts and the long exact homology sequence (A), we obtain the exactness of this lemma.

Theorem 2.5 Let C = (C"p in" I'] be an inverse system offree abelian chain complexes and let (C*(C'),d') be any strong chain complex. Then there exists a short exact sequence

in which ker(e,,_1) ==

EB

p+q=,,-1

and coker(e,,) ==

Tor(H;(C), Hq(C'))

EB H;(C) 0 Hq(C'),

p+q="

where e" : H,,+1(B- 0 C*(C')) --+ H,,(K 0 C*(C')) is the connecting homomorphism induced by the short exact sequence 0 --+ K 0 C*(C') ~ D* (C) 0 C*(C') ~ B- 0 C*(C') --+ 0 of chain complexes. Proof From the long exact homology sequence (A), we obtain the short exact sequence 0--+ coker(e,,) --+ H,,(D*(C) 0 C*(C')) --+ ker(e,,_t} --+ 0,

for each n E Z. We now show that the second term and the fourth one of the above exact sequence are isomorphic to the required forms. Consider the following short exact sequence

Tensoring with

Hq ( C'), we obtain. an exact sequence

and thus an isomorphism ker(i 01) == Tor(H;(C), Hq(C')). Taking the direct sum of this exact sequence over p+ q commutative diagram

= n, by Lemma 2.4 we have the following

H,,+1(B- 0 C*(C')) ~ H,,(K 0 C*(C'))

1==

ffi W B;+1 0 Hq(C')

1==

$(;01) -->

p+q="

ffi W

p+q="

$~1)

EB

p+q=n

tc,

0 Hq(C')

H;(C) 0 Hq(C') --+ 0

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ACTA MATHEMATICA SCIENTIA

Vo1.22 Ser.B

and thus the following isomorphisms: ker(Cn_l) ~ ker($(i@ 1)) ~ and coker(c n) ~

EB

EB

Tor(fI;(C), fIq(C'))

p+q=n-l

p+q=n

fI;(C) @ fIq(C')

which complete the proof of this theorem. Remark

It is worth while to note that the short exact sequence in Theorem 2.5 is split.

We get this result from the hypothesis of the inverse system of free abelian chain complexes. abelia~ chain complexes and the strong subchain complex (iJ.(C),d) of (C.(C),d) in Lemma 2.4 and Theorem 2.5 are really necessary. The fact that a direct product of free abelian groups need not be free abelian shows that those restrictions are needed. It seems natural to ask that the restrictions of the free

3

An Isomorphism on Strong (r-Stage) Homology

For a given chain complex (C.(C), d), we can construct another subchain complex (C:(r\C), d"(r»), r ~ 0, of (C.(C),d) whose p-th chain group is defined by r

c;(r)(c)

= II II (C-y)p+n, t = ('Yo,'Yl,"','Yn) n=OiEr n

and the boundary operator d;(r) : C;¥l(C)

-+

C;(r\C) is obtained by the restriction dplc.(r)(C)' p+l

p E Z. We have the p-th homology group fI;(r)(C),p E Z, of the subchain complex (C:(r)(C), d"(r») of (C.(C),d) and we also have the homomorphisms

and induced by the natural projections

and

i;# : Cp(C)

-+

c;(r)(c)

respectively. The p-th strong r-stage homology group fI~r)(C),p E Z, is defined by

In this case, we have an inverse system

No.1

Lee: ANALOGUE OF KUNNETH THEOREM IN STRONG HOMOLOGY

where J;~r+l : H~r+l)(C)

-+

39

H~r)(C) is a homomorphism and N is the set of all non-negative

integers. The derived limit functor lim" (-), the strong r-stage homology groups H~r) (C) and the strong homology groups Hp(C) of C are connected by the following two exact sequences[8,lO]:

and

(C) An inverse system 9 = (G-y, Yn', I") has the Mittag-Leffler property[ll] if every -y E admits a -y' E r, -y' 2: -y such that

Yn,(G-y') for any -y" 2: -y'. It is well known that if

r

= Yn,,(G-y")

9 has the Mittag-Leffler property, then

Definition 3.1 The dimension dim C of an inverse system C = (C-y, In" I'] is defined as the smallest positive integer m such that the n-th chain group (C-y)n is trivial for all n 2: m + 1 and -y E r. From this definition, we know that if dim C = m, then the ordinary homology group

Hn(C-y) of the chain complex C: is trivial for all n 2: m + 1 and -y E r. Theorem 3.2 Let C = (C-y'/n" I'] be an inverse system of chain complexes with dimension dim C = m 2: O. Then the homomorphism

induced by the projection is an isomorphism for r > m - p. Proof Since C = (C-y, In" I'] is an inverse system with dimension dim C = m, the homology group Hp+r(C-y) of C-y -is trivial for all p+r > m and -y E r. Thus we have the trivial inverse system Hp+r(C) = (Hp+r(C-y), I n ,*, r) = 0 of trivial groups Hp+r(C-y) = 0 and trivial homomorphisms I n ,* = O. So we also have

for all r

2: 0 and p + r > m. By (B), we have an isomorphism

Hence the inverse system (H;~l(C),j) has the Mittag-Leffler property. By this property and (C), we have and

40

ACTA MATHEMATICA SCIENTIA

for r

>m-

p.

Corollary 3.3

Let C

= (C"

Vo1.22 Ser.B

In" I'] be an inverse system of free abelian chain complexes

and let C' = (C~, I~..", I") be an inverse system of chain complexes with dimension dim C' Then, for r > m - q, there exists a short exact sequence

= m.

0-+ coker(c n ) -+ Hn(fJ.(C) @C.(C')) -+ ker(c n _ l ) -+ 0 in which

EB

ker(Cn _ l ) ~

Tor(fI;(C), fI~r)(c,))

p+q=n-l

and coker(cn ) ~

EB

fI;(C)

@

fI~r)(c,),

p+q=n

where Cn-l and Cn are the connecting homomorphisms as noted earlier. Proof By Theorem 2.5 and Theorem 3.2, we have the proof. I would like to thank the referee for the suggestions, especially a composition in an earlier version of Theorem 2.5. I want to express my gratitude to professors Hong-Jae Lee and Charles A. McGibbon for the insights which they have shared with me over the years. Acknowledgement

References 1 Bredon G E. Topology and geometry. New York: Springer-Verlag, 1993 2 Fuchs L. Infinite abelian groups. New York: Academic Press, 1970 3 Lee D W. Derived cup product and (strictly) derived groups. Bull Korean Math Soc, 1998,35(4):791-807 4 Lee D W. Strictly derived groups and t-step strong homology groups. Far East J Math Sci, 1998,6(3):401414 5 Lee H J, Lee D W. On the strong homology groups. Honam Math J, 1996,18(1):103-111 6 Lee H J, Lee D W. Strong homology groups w.r.t. L:PEz CP(X; R). Honam Math J, 1997,19(1): 131-138 7 Lisica J T, Mardesic S. Strong homology of inverse system of spaces I,n. Topology Appl, 1985,19: 29-64 8 Mardesic S, Miminoshvili Z. The relative homeomorphism and wedge axioms for strong homology. Glasnik Mat, 1990, 25(45): 387-416 9 Mardesi" S, Prasolov A V. Strong homology is not additive. Trans Arner Math Soc, 1988,307(2): 725-744 10 Mardesi" S, Prasolov A V. On strong homology of compact spaces. Topology Appl, 1998,82: 327-354 11 Mardesi" S, Segal J. Shape theory. New York, Amsterdam: North-Holland Publ Co, 1982 12 Mardesi" S, Watanabe T. Strong homology and dimension. Topology Appl, 1988,29:185-205 13 McGibbon CA. Phantom maps, Handbook of algebraic topology. New York: North-Holland, 1995 14 McGibbon C A, Steiner R. Some questions about the first derived functor of the inverse limit. J Pure Appl Algebra, 1995, 103: 325-340 15 Mdzinarishvili L, Spanier E. Inverse limits and cohomology. Glasnik Mat, 1993, 28(48): 167-176 16 Spanier E. Algebraic topology. New York: McGraw-Hill, 1966