UDC 513.836

MATHEMATICS

V. M. BUKHSHTABER

REPRESENTING SPACES FOR THE \(K\)-FUNCTOR WITH COEFFICIENTS

(Presented by Academician P. S. Aleksandrov on 26 IX 1968)

This paper describes the connection between the Adams–Toda homotopy operations (see \((^1)\), Theorem 1.7) and the differentials in the Atiyah–Hirzebruch spectral sequence (see \((^2)\)). The results obtained make it possible to describe the homotopy type of representing spaces for \(K\)-theory with coefficients in the rings \(Z_p\) and \(Z[t]/(t+1)^p-1\), where \(p\) is an odd prime.

We shall consider only complex \(K\)-theory. The Atiyah–Hirzebruch spectral sequence for a complex \(X\) will be denoted by \(SP(X)\) for the \(K\)-functor and by \(SP(X; Z_p)\) for the \(K\)-functor mod \(p\). We note that by a cellular complex we shall mean a cellular complex with finite skeleta.

I. Lemma 1. For any \(n>2p-3\) there exists a sequence of mappings
\[ A(n)=\{\alpha_s;\alpha_s:S_p^{n+2s(p-1)-1}\to S_p^n\} \]
such that: a)
\[ \alpha_s=(\Sigma^{2s(p-1)}\alpha_1)\circ\cdots\circ\alpha_1; \]
b) for the complex
\[ W_s=S_p^n\cup_{\alpha_s}D_p^{\,n+2s(p-1)}, \]
in the spectral sequences \(SP(W_s)\) and \(SP(W_s,Z_p)\) the differential \(d_{2s(p-1)+1}\) acts nontrivially. Here
\[ S_p^n=S^n\cup_p D^{n+1}. \]

Proof. Consider a mapping \(f:S_p^{2p}\to S^3\) with Hopf invariant \(1 \bmod p\) (see \((^3)\)). It is easy to show that for \(n>2p-3\) the mapping \(\Sigma^{n-3}f\) decomposes into the composition
\[ \pi\circ\alpha_1:S_p^{n+2p-3}\to S_p^{n-1}. \]
Since
\[ d_{2p-1}=e_{2,1}, \]
we have \(K^*(W_1)=0\), and the mapping
\[ \alpha_1^*:K^*(S_p^{n-1})\to K^*(S_p^{n+2p-3}) \]
is an isomorphism. Consequently, for every \(s\) the mapping \(\alpha_s^*\) is an isomorphism and \(K^*(W_s)=0\). The lemma is proved.

Theorem 1. For any \(n>2p-3\) there exists a cellular complex \(Y(Z_p,n)\) such that:

a)
\[ \pi_q(Y(Z_p,n))= \begin{cases} Z_p, & q\equiv n \pmod {2(p-1)},\\ 0, & q\not\equiv n \pmod {2(p-1)}; \end{cases} \]

b) the generating element
\[ u_n\in H^n(Y(Z_p,n),Z_p) \]
is a cycle of all differentials in \(SP(Y(Z_p,n);Z_p)\).

The complex \(Y(Z_p,n)\) has the following properties:

1) All cellular complexes \(Y\) satisfying conditions a) and b) are homotopy equivalent to the complex \(Y(Z_p,n)\).

2) A mapping
\[ f:Y(Z_p,n)\to Y(Z_p,n) \]
is a homotopy equivalence if and only if \(f^*(u_n)\ne0\).

Proof. From the sequence \(A(n)=\{\alpha_s\}\) one can construct a sequence of cellular complexes \(\{Y_s\}\) as follows. As \(Y_0\) take the Eilenberg–Mac Lane space \(K(Z_p,n)\), \(n>2p-3\); as \(Y_1\), take the space of the fibration
\[ Y_1 \xrightarrow{k(Z_p,n+2(p-1))} Y_0 \]
with Postnikov invariant \(e_{2,1}\). Denote by \(a_1\) the generator of the group \(H^n(Y_1;Z_p)\), and by \(b_1\) the generator of the group \(\pi_n^p(Y_1)\). By construction, in \(SP(Y_1;Z_p)\) the element
\[ a_1\in E_{2p-1}^{n,2t} \]
is a cycle of the differential \(d_{2p-1}\), and therefore the generator of the group \(\pi_{n+2(p-1)}(Y_1)\) can be decomposed into the composition
\[ S^{n+2(p-1)}\subset S_p^{n+2(p-1)} \xrightarrow{\alpha_1} S_p^n\subset Y_1. \]
According to Theorem 2 of \((^5)\), the element
\[ a_1\in E_{2p-1}^{n,2t} \]
is a cycle of the differentials \(d_r\) for \(r<4(p-1)+1\)

The embedding \(b_1:S_p^n\subset Y_1\) extends to a map \(\tilde b_1:W_2\to Y_1\), and since, by Lemma 1, the differential \(d_{4(p-1)+1}\) acts nontrivially in the complex \(W_2\), we obtain that \(d_{4(p-1)+1}(a_1)\ne 0\). Thus, in the group
\[ E_{4(p-1)+1}^{\,n-4(p-1)+1,\;2t-4(p-1)}\subset SP(Y_1,Z_p) \]
an element is marked. Take as \(Y_2\) the fibration space
\[ Y_2 \xrightarrow{K(Z_p,\;n+4(p-1))} Y_1 \]
with Postnikov invariant \(c_2\in H^{n+4(p-1)+1}(Y_1,Z_p)\), where \(c_2\) is some representative of the element \(d_{4(p-1)+1}(a_1)\) in cohomology. Denote by \(a_2\in H^n(Y_2,Z_p)\) and \(b_2\in \pi_n^p(Y_2)\) the generating elements. Using the complex \(W_2\), we obtain that the generating group \(\pi_{n+4(p-1)}(Y_2)=Z_p\) decomposes into the composition
\[ S^{n+4(p-1)}\subset S_p^{n+4(p-1)} \xrightarrow[\ b_2\ ]{a_2} S_p^n\subset Y_2. \]
Continuing the construction, we obtain a sequence of complexes
\[ \{\,Y_i,\; Y_i \xrightarrow{K(Z_p,\;n+2i(p-1))} Y_{i-1}\,\},\quad i\ge 0. \]
The \((n+2i(p-1)-1)\)-dimensional skeleta of the spaces \(Y_i\) and \(Y_{i-1}\) can be identified for any \(i\), and therefore we have a sequence of embedded cell complexes
\[ [Y_0]_{n+2(p-1)-1}\subset [Y_1]_{n+4(p-1)-1}\subset\cdots\subset [Y_i]_{n+2(i+1)(p-1)-1}\subset\cdots . \]
Put
\[ Y(Z_p,n)=\lim_{\longrightarrow}[Y_i]_{n+2(i+1)(p-1)-1}. \]
It is clear that \(Y(Z_p,n)\) is a cell complex. Denote by \(u_n\in H^n(Y(Z_p,n),Z_p)\) a generator of the group.

By construction, \(u_n\) is a cycle of all differentials in \(SP(\ ,Z_p)\), and moreover, for any cell complex \(X\) and element \(x\in H^n(X,Z_p)\) that is a cycle of all differentials in \(SP(\ ,Z_p)\), there exists a continuous map \(f:X\to Y(Z_p,n)\) such that \(f^*u_n=x\). Using now the fact that, for any \(s\), the generator of the group \(\pi_{n+2s(p-1)}(Y(Z_p,n))\) decomposes into the composition
\[ S^{n+2s(p-1)}\to S_p^{\,n+2s(p-1)} \xrightarrow{\alpha_s} S_p^n\subset Y(Z_p,n), \]
we obtain the proof of properties 1) and 2) of the space \(Y(Z_p,n)\). The theorem is proved.

Since the spectral sequence \(SP(Y,(Z_p,n)Z_p)\) converges strongly \((^4)\), in the group \(k^n(Y(Z_p,n)Z_p)\) there exists an element \(\zeta_n\) of filtration \(n\).

Lemma 2. a) For any \(n>2p-3\) there exists a homotopy equivalence
\[ \chi_n:\Omega Y(Z_p,n+1)\xrightarrow{\simeq}Y(Z_p,n). \]
b) There exists a homotopy equivalence
\[ \chi:\Omega^{2(p-1)}Y(Z_p,2(p-1))\to Z_p\times Y(Z_p,2(p-1)). \]

Proof. Let \(K_p\) be a representing space for the \(K\)-functor \(\bmod p\). Denote by \(K_p(2n)\) the \(2n\)-decreasing space for the complex \(K_p\), and by \(K_p(2n-1)\) the space \(\Omega K_p(2n)\). According to the classifying properties of the spaces \(K_p(n)\), there exists a map \(f:Y(Z_p,n+1)\to K_p(n+1)\) such that \(f^*(\eta_{n+1})=\zeta_{n+1}\), where \(\eta_{n+1}\) is the universal element. Consider the map
\[ \Omega f:\Omega Y(Z_p,n)\to \Omega K_p(n). \]
Since the homomorphism
\[ (\Omega f)^*:H^n(\Omega K_p(n+1),Z_p)\to H^n(\Omega Y(Z_p,n+1),Z_p) \]
is nonzero, the generating element
\[ c\in H^n(\Omega Y(Z_p,n+1),Z_p) \]
is, obviously, a cycle of all differentials in \(SP(\ ,Z_p)\). Applying now Theorem 1, we obtain that the spaces \(\Omega Y(Z_p,n+1)\) and \(Y(Z_p,n)\) are homotopy equivalent. Part a) is proved. Part b) is proved analogously.

Corollary 1. For any \(n>2p-3\) there exists a homotopy equivalence \(\lambda\):
\[ \prod_{k=0}^{p-1} Y(Z_p,n+2k)\to K_p(n). \]

Put
\[ h_p^{-q}(X,Y)= [S^q(X/Y),\, Z_p\times Y(Z_p,2(p-1))]_*, \quad q>0. \]
The homotopy equivalence
\[ \chi:\Omega^{2(p-1)}Y(Z_p,2(p-1))\to Z_p\times Y(Z_p,2(p-1)) \]
makes it possible to define the groups \(h_p^{-q}\) also for negative-

\(q\). By the standard method it is easy to verify that the homotopy functor \(h_p^*\) defines an extraordinary cohomology theory \((2(p-1))\)-periodic). Applying Lemma 2 and Corollary 1, we obtain the theorem:

Theorem 2. On the category of cellular complexes there exists a \(2(p-1)\)-periodic cohomology theory \(h_p^*=\sum h_p^q\) such that for any pair of cellular complexes \((X,Y)\) there is a natural isomorphism

\[ \tau_p:\ \sum_{i=0}^{p-1} h_p^{-2i}(X,Y)\ \xrightarrow{\ \approx\ }\ k^0(X,Y;\mathbb Z_p). \]

II. Let \(L_p^\infty\) be the infinite-dimensional lens space. Denote by \(L^n\) the \(2n\)-dimensional skeleton of the space \(L_p^\infty\). Denote the group \(k^0(L^n)\) by \(A_n\) and put, as usual, \(\widetilde{k}^0(X;A_n)=k^0(X\# L^n)\). Since \(L^1=S_p^1\), it follows that \(k^*(\,;\,A_1)=k^*(\,;\mathbb Z_p)\). For any \(n\) the equality \(L^{n+1}/L^n=S_p^{2n+1}=S^{2n}S_p^1\) holds, which makes it possible to generalize Theorem 2.

Theorem 3. On the category of cellular complexes, for every \(n>0\) there exist \(2(p-1)\)-periodic cohomology theories \(h_{p,n}^*\) and natural transformations \(\tau_{p,n}^0:h_{p,n}^0\to k^0(\,;\,A_n)\), satisfying the following conditions:

a) for any pair of cellular complexes there is an isomorphism

\[ \tau_p:\ \sum_{i=0}^{p-1} h_{p,n}^{-2i}(X,Y)\to k^0(X,Y;A_n), \]

b) there are transformations \(p_n:h_{p,n}\to h_{p,n-1}\) such that \(\pi_n^*\tau_{p,n}=\tau_{p,n-1}p_n\), where the projection \(\pi_n:k^*(\,;\,A_n)\to k^*(\,;\,A_{n-1})\) is induced by the inclusion \(\pi_n:L^{n-1}\to L^n\).

Proof. Denote by \(V^n\) the space of continuous mappings \(\{L^n\to BU\}_*\) preserving the base point. Since \(L^n\) is a finite complex, the space \(V^n\) is a representing space for the \(k^*(\,;\,A_n)\)-theory. Suppose that for all \(q\le n\) cellular complexes \(Y_q\) and mappings \(p_q:Y_q\to Y_{q-1}\), \(\tau_q^0:Y_q\to V^q\) have been constructed such that \(\tau_{q-1}^0p_q\simeq f_{q-1}\tau_q^0\) and \(\tau_q^0\) induces an isomorphism of the groups \(\pi_{2s(p-1)}\), \(s\ge 0\). (For \(n=1\), \(V^1=K_p\), therefore as \(Y_1\) one may take \(Y(\mathbb Z_p,2(p-1))\).) We construct the complex \(Y_{n+1}\) and mappings \(p_n\) and \(\tau_{n+1}^0\) satisfying these conditions. Consider the segment of the Puppe sequence

\[ (\pi_n:L^n\subset L^{n+1}):\ S^{2n+1}S_p^1\to S^2L^n\to S^2L^{n+1}. \]

There is a Serre fibration \(g:\Omega^2V^n\to \Omega^{2n+1}K_p\) with fiber \(F^a\simeq\Omega^2V^{n+1}\), and moreover \(i=\Omega^2 f_n\cdot a\), where \(i:F\to\Omega^2V^n\) is the inclusion of the fiber. Using standard homotopy technique, it is easy to show that \(F\) is homotopy equivalent to the fibration space \(F'\to\Omega^2V^n\), induced by the universal Serre fibration \(*=p\Omega^{2n+1}K_p*\to\Omega^{2n+1}K_p\) by means of the mapping \(g\). Consider the composition of mappings

\[ \varphi:\ \Omega^2(Y_n)_0 \xrightarrow{\tau_n^0} \Omega^2(V^n)_0 \xrightarrow{g} \Omega^{2n+1}(K_p)_0 \xrightarrow{\Omega^{2n+1}} \Omega^{2n+1}Y(\mathbb Z_p,2p-1) \]

and denote by \(Y_{n+1}\) the fibration space \(Y_{n+1}\to\Omega^2(Y_n)_0\), induced by the fibration \(*\to\Omega^{2n+1}Y(\mathbb Z_p,2p-1)\) by means of the mapping \(\varphi\). It follows directly from the construction that there exist mappings \(p_n\) and \(\tau_{n+1}^0\) satisfying the required conditions. The induction step is complete.

Put \(h_{p,n}^{-q}(X,Y)=[S^qX/Y,A_n\times Y]\). It is easy to verify that the cohomology theory \(h_{p,n}^*\) and the transformation \((\tau_n^0)^*:h_{p,n}=k^0(\,;\,A_n)\) satisfy all the conditions of the theorem.

III. We now consider the \(K\)-functor with coefficients in the ring \(k^*(L_p^\infty)\). Since \(H^*(L_p^\infty;Q)=0\), we have \(k^*(L_p^\infty)=\mathcal K^*(L_p^\infty)\) (see (4)). It is easy to show that \(\mathcal K^1(L_p^\infty)=0\) and \(\mathcal K^0(L_p^\infty)=\mathbb Z[t]/(t+1)^p-1\), where \(t=\zeta-1\), and \(\zeta\) is the canonical line bundle over \(L_p^\infty\). Denote the ring

\(\mathbb Z[t]/(t+1)^p-1\) by \(A\). The ring \(A\) is additively isomorphic to the direct sum of \((p-1)\) copies of the ring of integral \(p\)-adic numbers, and therefore in questions where consideration of the \(K\)-functor with \(p\)-adic coefficients is required, one may use the functor \(k(\ ; A)\) (see \((^4)\), § 8). In \((^4)\) it is shown that if \(X\) is a finite complex, then there is an isomorphism
\[ k^*(X; A) \simeq K^*(X)\otimes_{\mathbb Z} A. \]
For an arbitrary complex \(X\) there exists an exact sequence relating the groups \(k^*(X; A)\) and \(k^*(X)\). From the results of Section II it follows:

Theorem 4. On the category of cellular complexes there exists a \(2(p-1)\)-periodic cohomology theory \(\mathscr H_p\) and a natural transformation \(\tau_p^0:\mathscr H_p^0\to k^0(\ ; A)\), which for any pair of complexes \((X,Y)\) induces an isomorphism
\[ \tau_p:\ \sum_{i=0}^{p-1}\mathscr H_p^{-2i}(Z,Y)\to k^0(X;A). \]

Proof. From Theorem 3 it follows that for any pair of cellular complexes \((X,Y)\) and any integer \(q\) an inverse sequence of groups \(\{h_{p,n}^q(X,Y)\}_{n>0}\) is defined. Since \(H^*(L^n;Q)=0\), for any cellular complex \(X\) there is an isomorphism
\[ k^q(X;A_n)\simeq \lim_{\leftarrow} k^q(X_l;A_n), \]
where \(X_l\) is the \(l\)-dimensional skeleton of the complex \(X\). For every \(l\) the group \(k^q(X_l;A_n)\) is finite; hence the group \(k^q(X;A_n)\) is profinite. It follows that the group \(h_{p,n}^q(X)\subset k^q(X;A_n)\) is also profinite. Put
\[ \mathscr H_p^q(X,Y)=\lim_{\leftarrow} h_{p,n}^q(X,Y). \]
Using \((^4)\), § 3, we obtain that the functor \(\mathscr H_p^*=\sum \mathscr H_p^q\) defines a cohomology theory on the category of cellular complexes. For any complex \(X\) there is an isomorphism
\[ k^*(X \# L_p^\infty)\simeq K^*(X \# L_p^\infty) \]
(see \((^4)\)). Clearly,
\[ k^*(X \# L_p^\infty)\simeq \lim_{\leftarrow} k^*(X \# L^n). \]
Thus, the sequences of transformations \(\{\tau_{p,n}^0\}\) and \(\{\tau_{p,n}\}\) induce transformations \(\tau_p^*:\mathscr H_p^0\to k^0(\ ; A)\) and
\[ \tau_p:\ \sum \mathscr H_p^{-2i}(X,Y)\to k^0(X;A). \]
The theorem is proved.

IV. In Section II we introduced a sequence of fibrations in the sense of Serre
\[ \{f_n: V\to V^n,\ f_n^{-1}(*)\simeq \Omega^{2n}K_p\}. \]
Put \(V=\lim_{\leftarrow}V^n\). Clearly, for every \(n\) the projection \(\pi_n:V\to V^n\) is a fibration in the sense of Serre. All \(V^n\) are \(H\)-spaces and \(f_n\) are homomorphisms; therefore \(V\) is also an \(H\)-space.

Theorem 5. For every finite complex \(X\) there is an isomorphism \(k^0(X;A)=[X,V]_*\), where \(A=k^*(L_p^\infty)\).

Proof. It is easy to verify that there is an epimorphism
\[ \alpha:\ [X,V]_*\to \lim_{\leftarrow}[X,V^n]=k^0(X,A). \]
Introduce in the space \(V\) the filtration \(\{V_n\}\), putting
\[ V_0=V,\qquad V_n=\pi_n^{-1}(*),\quad *\in V^n. \]
Denote by \(i_n:V_n\subset V\) and \(i_{n,k}:V_n\subset V_k\) the natural embeddings. Clearly, all \(V_n\) are fibrations in the sense of Serre.
\[ V_n=\lim_{\leftarrow} V_n^k. \]
Let
\[ \pi_n^k:V_n\to V_n^k \]
be the corresponding projections of fibrations.

Lemma 3. Let \(X\) be a cellular complex and let \(\{\varphi_n:X\to V_n\}\) be a sequence of maps such that \(\varphi_n\simeq i_{n+1,n}\circ\varphi_{n+1}\). Then the map \(\varphi=\varphi_0:X\to V\) is homotopic to a map to a point.

Now let \(X\) be a finite complex and \(x\in\ker\alpha\). Put \(\varphi_0=x\), and, using the fact that for every finite complex the group \([X,\Omega^{2n}K_p]\) is finite, we can by induction construct a sequence of maps satisfying the lemma. The theorem is proved.

Mechanical-Mathematical Faculty
of Moscow State University
named after M. V. Lomonosov

Received
20 IX 1968

References

1. J. Adams, Mathematics Translation Collection, 12, 3 (1968).
2. M. Atiyah, F. Hirzebruch, ibid., 6, 2 (1962).
3. J.-P. Serre, Collected Works: Fibre Spaces, Moscow, 1958.
4. M. E. Klicheber, A. S. Mishchenko, News of the Academy of Sciences of the USSR, Mathematical Series, 32, No. 3 (1968).
5. V. M. Buchstaber, Mathematical Collection, No. 2 (1969).