MATHEMATICS

M. BOKSHTEIN

TENSOR PRODUCTS OF SYSTEMS OF GROUPS AND UNIVERSAL COEFFICIENT THEOREMS FOR HOMOLOGY AND COHOMOLOGY

(Presented by Academician P. S. Aleksandrov, 18 XII 1957)

1. The problem of finding a universal coefficient group for the spectral groups of homology or cohomology of a topological space in the sense of Čech was the subject of a paper by Steenrod (¹). However, owing to the use of an incorrect algebraic lemma, what is actually obtained in that paper is not the formula

\[ H^{q}(X,G)\approx H_0^q(X)\otimes G+H_0^{q+1}(X)*G, \tag{1} \]

which would mean the universality of the integral coefficients for such cohomology groups, but only the weaker fact that the sequence

\[ 0\to H_0^q(X)\otimes G\to H^q(X,G)\to H_0^{q+1}(X)*G\to 0 \tag{2} \]

is exact, i.e. that \(H^q(X,G)/H_0^q(X)\otimes G\approx H_0^{q+1}(X)*G\). Here \(H^q(X,G)\) is the \(q\)-dimensional spectral cohomology group of the space \(X\) with coefficient group \(G\); \(H_0^q(X)=H^q(X,I)\), where \(I\) is the additive group of integers; \(\otimes\) is the tensor product sign; \(*\) is the torsion product sign. Eilenberg and Mac Lane were able to correct the proof of formula (1), but only for metric compacta ((²), the note on p. 820 and Theorem 44.2).

The universality of the group \(I\) in the general case was proved by me in the paper (³) (Chapter I, § 5), but without an explicit indication of the formula giving the group \(H^q(X,G)\). The results of that paper were obtained during the 1940s (the paper was completely written by 1950), and therefore modern methods were not used in it; from their point of view the proof given there, after the introduction of the corresponding notions, becomes considerably shorter and more transparent. At the same time, as we shall now show, in the general case one can prove formula (1).

2. We introduce the following definitions. By a system of groups \(\{G_\alpha;\pi\}\) we shall mean a collection of additive Abelian groups \(G_\alpha\), for some of which homomorphisms \(\pi=\pi_\beta^\alpha\) of the group \(G_\alpha\) into the group \(G_\beta\) are defined (in particular, if the indices \(\alpha\) are directed partially ordered, then the system is called a direct, inverse, or two-sided spectrum when these homomorphisms are defined, respectively, for \(\alpha<\beta\), \(\alpha>\beta\), and in both cases). Two systems \(\{G_\alpha;\pi\}\) and \(\{H_\alpha;\omega\}\) are called conjugate if they are defined for the same set of indices and if the homomorphisms \(\omega_\alpha^\beta\) are defined if and only if the homomorphisms \(\pi_\beta^\alpha\) are defined. The tensor product of two conjugate systems

\[ \{G_\alpha;\pi\}\otimes\{H_\alpha;\omega\} \]

is called the factor group of the direct sum

$\sum_\alpha G_\alpha \otimes H_\alpha$ by the subgroup generated by all elements of the form $g_\alpha \otimes \omega^\beta_\alpha h_\beta - \pi^\alpha_\beta g_\alpha \otimes h_\beta$ $(g_\alpha \in G_\alpha,\ h_\beta \in H_\beta)$.

We have proved a theorem ((3), Chap. I, § 3, or (4)), which, with the aid of the definition introduced, can be written in the form

\[ H^q(X,G) \simeq \{H^q_m(X);\ \pi^{m'}_m,\ \omega^m_{m'}\}\otimes \{\,_mG;\ i^m_{m'},\ j^{m'}_m\}, \tag{3} \]

where $m$ runs through the nonnegative integers; $H^q_m(X)=H^q(X,I_m)$; $I_0=I$; $I_m=I/mI$; $mI$ is the subgroup of elements of the group $I$ divisible by $m$ (similarly, in general, $mG$ and $G_m$ are defined for an arbitrary group $G$); $_mG$ is the subgroup of elements $g\in G$ for which $mg=0$; the homomorphisms $\pi^{m'}_m,\ \omega^m_{m'},\ i^m_{m'},\ j^{m'}_m$ are defined for $m\mid m'$; $i^m_{m'}$ is the embedding map of the subgroup $_mG$ into the subgroup $_{m'}G$; $j^{m'}_m$ is the map of $_{m'}G$ into $_mG$ arising as a result of multiplication of the elements of the group $G$ by the number $\frac{m'}{m}$; $\pi^{m'}_m$ is a homomorphism of $H^q_{m'}(X)$ into $H^q_m(X)$, generated by reducing the coefficient group modulo $m$; $\omega^m_{m'}$ is a homomorphism of $H^q_m(X)$ into $H^q_{m'}(X)$, generated by the map $I_m$ into $I_{m'}$ induced by multiplication by the number $\frac{m'}{m}$. Since the system $\{\,_mG;\ i^m_{m'},\ j^{m'}_m\}$ is determined by specifying the group $G$, formula (3) shows that the system $\{H^q_m(X);\ \pi^{m'}_m,\ \omega^m_{m'}\}$, which we call the modular spectrum of the cohomology groups of the space, is a universal system of cohomology groups, i.e. determines the cohomology groups of the space for any coefficient group.

1. We recall the proof of formula (3). Let $\sum u_i\otimes g_i$ be an element of the tensor product of the right-hand side of this formula $(u_i\in H^q_{m_i}(X),\ g_i\in\,_{m_i}G,\ i=1,\ldots,s)$. To it there corresponds uniquely an element (which we shall denote by $\sum g_i u_i$) of the group $H^q(X,G)$ containing the cocycle $l^q=\sum g_i l^q_i$, where $l^q_i$ are integral cochains of some covering $\Omega^\alpha$ of the space $X$ which, after reduction modulo $m_i$, will be cochains from the cohomology classes $u_i$. Conversely, let $u\in H^b(X,G)$, and let $l^q$ be a cocycle, contained in $u$, of some covering $\Omega^\alpha$ of the space $X$. As is known, for every finite complex, and hence also for the nerve of the covering $\Omega^\alpha$, there exist such free generators $l^q_i$ of the groups of integral cochains (the values assumed by the index $i$ for different values of the dimension $q$ overlap only partially) that $\nabla l^q_i=m_{iq}l^{q+1}_i$, where $\nabla$ is the coboundary operator, and $m_{iq}$ are nonnegative integers (depending on $i$ and $q$). Therefore $l^q=\sum g_i l^q_i$, $g_i\in G$, $\nabla l^q=m_{iq}g_i l^{q+1}_i=0$ and, hence, $m_{iq}g_i=0$, i.e. $g_i\in\,_{m_{iq}}G$, and consequently the element $u$ always corresponds to some element $\sum u_i\otimes g_i$ of the tensor product from formula (3). Finally, if $\sum \bar g_k u_k=0$, $\bar g_k\in\,_{\bar m_k}G$, $\bar u_k\in H^q_{\bar m_k}(X)$, then in some covering $\Omega^\alpha$ we have $\sum \bar g_k \bar l^q_k\sim 0$, $\bar l^q_k\in \bar u_k$, i.e., decomposing with respect to the generators $l^q_i$ of the above-mentioned basis (constructed for $\Omega^\alpha$),

\[ \sum_{i,k}\bar g_k \lambda_{ik} l^q_i=\nabla\left(\sum_i g'_i l^{q-1}_i\right)=\sum_i m'_i g'_i l^q_i, \]

where $g'_i\in G$, whence $\sum_k \bar g_k\lambda_{ik}=m'_i g'_i$. On the other hand, the relation

\[ \nabla \bar l^q_k=0,\quad \text{i.e.}\quad \nabla\left(\sum_i \lambda_{ik}l^q_i\right)=\sum_i \lambda_{ik}m_i l^{q+1}_i\equiv 0\pmod{\bar m_k} \]

gives $m_i\lambda_{ik}=0\pmod{\bar m_k}$, i.e. $\lambda_{ik}=\lambda'_{ik}\bar m_k/\delta_{ik}$, where $\lambda'_{ik}$ are integers and $\delta_{ik}=(m_i,\bar m_k)$, and therefore

\[ \sum_k \bar u_k \otimes \bar g_k = \sum_{i,k} \lambda'_{ik}\omega^{\frac{\delta_{ik}}{m_k}}\pi_{\delta_{ik}}^{m_i}u_i \otimes \bar g_k = \sum_{i,k} u_i \otimes \lambda'_{ik}\frac{\delta_{ik}\bar m_k}{m_i!\,\delta_{ik}!}\bar g_k = \sum_{i,k} u_i \otimes \lambda'_{ik}\frac{\bar m_k}{\delta_{ik}}\bar g_k = \sum_{i,k} u_i \otimes \lambda_{ik}\bar g_k = \sum_i u_i \otimes m'_i g'_i = 0, \]
i.e. the correspondence between the left- and right-hand sides of formula (3) will be one-to-one (here \(m_i=m_{i,q}\), \(m'_i=m_{i,q-1}\), \(m_i m'_i=0\), \(u_i\) is a cohomology class containing the cochain \(l_i^q\) reduced modulo \(m_i\)).

1. Let us note that if \(G\) is a ring, then formula (3) makes it possible to define the cohomology ring of the space with coefficients in \(G\) from its cohomology rings \(\bmod\, m\) (\(m=0,1,2,\ldots\)) and the mappings \(\pi\) and \(\omega\). For this, as is not hard to see, it is enough to define multiplication in the right-hand side of this formula as follows \((u_1\in H_{m_1}^{q_1}(X),\ u_2\in H_{m_2}^{q_2}(X),\ g_1\in_{m_1}G,\ g_2\in_{m_2}G)\):
   \[ (u_1\otimes g_1)\cdot(u_2\otimes g_2) = (\pi_\delta^{m_1}u_1\cdot \pi_\delta^{m_2}u_2)\otimes(g_1g_2), \]
   where \(\delta=(m_1,m_2)\), which makes sense, since, evidently, \(g_1g_2\in_\delta G\), i.e. \(\delta g_1g_2=0\) (for \(\delta=\lambda_1m_1+\lambda_2m_2\), where \(\lambda_1\) and \(\lambda_2\) are integers).

2. The universality of the group \(I\) for the spectral cohomology groups of a topological space can now be proved as follows. First we show without difficulty that the sequences
   \[ \cdots \to H_0^q(X)\xrightarrow{m}H_0^q(X)\xrightarrow{\pi}H_m^q(X)\xrightarrow{\chi}H_0^{q+1}(X)\xrightarrow{m}H_0^{q+1}(X)\to\cdots, \tag{4} \]
   \[ \cdots \to H_n^q(X)\xrightarrow{\omega}H_{mn}^q(X)\xrightarrow{\pi}H_m^q(X)\xrightarrow{\pi\chi}H_n^{q+1}(X)\xrightarrow{\omega}H_{mn}^{q+1}(X)\to\cdots, \tag{5} \]
   where \(m\) denotes the endomorphism consisting in multiplication of all elements of the group by the number \(m\), \(\chi=\chi_0^m\) is the boundary homomorphism introduced by us in \((9)\) (i.e. \(\chi_0^m u\), where \(u\in H_m^q(X)\), is the \((q+1)\)-dimensional integral cohomology class containing the cocycle \(\frac1m\nabla l\), where \(l\), after reduction modulo \(m\), belongs to \(u\)), and \(\pi\chi\) is the superposition \(\pi_n^0\chi_0^m\), are exact ((3), Ch. I, § 5, Lemma 2). Sequence (4), evidently, gives rise to a new exact sequence \((m\ne0)\)
   \[ 0\to [H_0^q(X)]_m \to H_m^q(X)\to {}_m[H_0^{q+1}(X)]\to0, \tag{6} \]
   where, as always, \([H_0^q(X)]_m=H_0^q(X)/mH_0^q(X)\).

Since all elements of the group \({}_m[H_0^{q+1}(X)]\) have orders bounded by the same number \(m\), by Prüfer’s theorem ((5), p. 156), \({}_m[H_0^{q+1}(X)]\) decomposes into a direct sum of cyclic groups. We shall show that one can construct a mapping
\[ \tilde\chi=\tilde\chi_0^m:\ {}_m[H_0^{q+1}(X)]\to H_m^q(X), \]
taking for each of the generating cyclic summands of the group \({}_m[H_0^{q+1}(X)]\) one of the elements of the group \(H_m^q(X)\) which maps into it under the mapping \(\chi\). Then \(\chi\tilde\chi\) is the identity mapping, and therefore the exact sequence (6) splits, i.e.
\[ H_m^q(X)\simeq [H_0^q(X)]_m+{}_m[H_0^{q+1}(X)]. \tag{7} \]

The arbitrariness present in the construction of the mapping \(\tilde\chi\) can be used in such a way as to obtain a kind of coherence of such direct decompositions. Namely, for each prime number \(p\), first in the indicated way we construct a mapping
\[ \tilde\chi_p^0:\ {}_p[H_0^{q+1}(X)]\to H_p^q(X), \]
and then we carry out the further construction inductively: if \(\tilde\chi_{p^{k-1}}^0\) has already been constructed, then for those generators \(a_i\) of the cyclic summands of the group \({}_{p^k}[H_0^{q+1}(X)]\) whose orders \(\le p^{k-1}\), we set
\[ \tilde\chi_{p^k}^0 a_i=\omega_{p^k}^{p^{k-1}}\tilde\chi_{p^{k-1}}^0 a_i, \]
and for the generators of order \(p^k\) we take as \(\tilde\chi_{p^k}^0 a_i\) one of the elements \(u\) of the group \(H_{p^k}^q(X)\), for

for which \(\pi_{p^{k-1}}^{p^k}u=\widetilde\chi_{p^{k-1}}^0(pa_i)\), \(\chi_0^{p^k}u=a_i\).} Since

\[ \pi_0^{p^{k-1}}\chi_0^{p^k}\widetilde\chi_{p^{k-1}}^0(pa_i)=\pi_0^0(pa_i)=0, \]

then, by virtue of the exactness of sequence (5) for \(m=p^{k-1}\), \(n=p\), there exists an element \(u'\) satisfying the first of these conditions. Then

\[ p(a_i-\chi_0^{p^k}u')=pa_i-\chi_0^{p^{k-1}}\pi_{p^{k-1}}^{p^k}u' =pa_i-\chi_0^{p^{k-1}}\widetilde\chi_{p^{k-1}}^0(pa_i)=pa_i-pa_i=0, \]

and therefore, by the exactness of sequence (4), \(a_i-\chi_0^{p^k}u'=\chi_0^p w\), and the element \(u=u'+\omega_{p^k}^p w\) will also satisfy the second condition (here the obvious formulas are used
\(\chi_0^m=\chi_0^{m'}\omega_{m'}^m,\ \dfrac{m'}{m}\chi_0^{m'}=\chi_0^{m-m'}\pi_m^{m'}\ (m\mid m')\)).

For the mapping \(\widetilde\chi_{p^k}^0\) thus constructed we shall have
\(\chi_0^{p^k}\widetilde\chi_{p^k}^0 a_i=a_i\), i.e. \(\chi\widetilde\chi=1\). If \(m=m_1\cdots m_n\), where \(m_i=p_i^{k_i}\) and \(p_1,\ldots,p_n\) are distinct primes, then we put
\[ \widetilde\chi_m^0=\sum_i \omega_m^{m_i}\widetilde\chi_{m_i}^0. \]
Then, as is easy to see, \(\chi_0^m\widetilde\chi_m^0=1\) will hold for all \(m\).

Again using, in an analogous way, the exactness of sequence (5), we obtain without difficulty that
\[ \pi_m^{m'}\chi_{m'}^0 a=\widetilde\chi_m^0\left(\frac{m'}{m}a\right),\qquad \omega_{m'}^{m''}\chi_{m'}^0 a=\widetilde\chi_{m''}^0 a\,(m/m'\!/m'',\ a\in {}_m[H_0^{q+1}(X)]). \]
This defines the homomorphisms \(\pi\) and \(\omega\) for the second summand of formula (7):
\[ \pi_m^{m'}=j_m^{m'},\qquad \omega_{m''}^{m'}=i_{m'}^{m''}. \]
For the first summand, however, which is a natural subgroup of the left-hand side of formula (7), \(\pi_m^{m'}\) coincides with the embedding \(\varphi_m^{m'}\) of the residue class mod \(m'\) into the residue class mod \(m(m/m')\) containing it, and \(\omega_{m''}^{m'}\) with \(\psi_{m''}^{m'}(m'/m'')\)—the passage from the residue class mod \(m'\) to the residue class mod \(m''\) as a result of multiplication by the number \(m''/m'\). Thus the modular spectrum of the cohomology groups of the space \(X\) is determined, and therefore formula (3) gives the group \(H^q(X,G)\):

\[ H^q(X,G)\approx \bigl([H_0^q(X)]_m+{}_m[H_0^{q+1}(X)];\pi,\omega\bigr)\otimes({}_mG;i,j)\approx \]

\[ \approx \bigl([H_0^q(X)]_m;\varphi,\psi\bigr)\otimes({}_mG;i,j) + \bigl({}_m[H_0^{q+1}(X)];j,i\bigr)\otimes({}_mG;i,j). \]

6. It is easy to see that
\[ \bigl([H_0^q(X)]_m;\varphi,\psi\bigr)\otimes({}_mG;i,j)\quad (m=0,1,2,\ldots) \]
coincides with \(H_0^q(X)\otimes G\). By virtue of the exactness of sequence (2) ((2), Theorem 40.3), it follows from this that
\[ \bigl({}_m[H_0^{q+1}(X)];j,i\bigr)\otimes({}_mG;i,j)\approx H_0^{q+1}(X)*G, \]
which can also be proved directly from the definition of the torsion product (here already \(m\ne0\), i.e. \(m=1,2,\ldots\)). This gives formula (3).

Everything that has been said is also applicable to the relative cohomology groups of \(X \bmod A\) \((A\subset X)\) and to the inner cohomology groups in the sense of P. S. Alexandrov (which take into account the noncompactness of the elements of a covering, \((^6)\)). In an analogous way, formula (3) (with \(q+1\) replaced by \(q-1\)) can be proved for singular homology groups of a space, which, however, was already known earlier, since it follows from general algebraic results ((\(^7\)), Ch. VI, Theorem 3.3). For singular cohomology groups and for spectral homology groups, formula (3) (and even the exact sequence (2)) does not hold in general, as is shown by counterexamples in \((^8,^1)\).

Received
17 XII 1957

REFERENCES

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