MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR S. P. NOVIKOV

RINGS OF OPERATIONS AND ADAMS-TYPE SPECTRAL SEQUENCES IN EXTRAORDINARY COHOMOLOGY THEORIES.

\(U\)-COBORDISMS AND \(k\)-THEORY

I. Let \(X=(X_n)\) be a spectrum of \((n-1)\)-connected complexes and maps \(f_n:\ EX_n\to X_{n+1}\) which are homotopy equivalences up to large dimension. \(X^*(K,L)\) denotes the cohomology of the pair \((K,L)\) with values in \(X\). By \(H^i(X,Z)\) we denote, naturally, \(\lim_{n\to\infty} H^{n+i}(X_n,Z)\), and by

\[ X^*(X)=\lim_{n\to\infty} X^i(X_n), \qquad \text{where } X^*(X)=\sum_i X^i(X), \]

the Steenrod ring \(A^X\).

One may consider the \(A^X\)-modules of cohomology

\[ X^*(K)=\sum_i X^i(K). \]

By analogy with the work \((^2)\), one can sometimes construct a spectral sequence \((E_r,d_r)\), where

\[ E_2=\operatorname{Ext}_{A^X}\bigl(X^*(K),X^*(L)\bigr). \]

The question arises: when does this spectral sequence converge to \(\operatorname{Map}^s[L,K]\)? We shall be interested in this question for \(L=S^0\).

Theorem 1. If for a point \(P\) the group \(X^0(P)=Z\), the cohomology \(H^*(X,Z)\) has no torsion, and the spectral sequence \((E_r,d_r)\) with term

\[ E_2=H^*(K,X^*(P)), \]

converging to \(X^*(K)\), has all differentials \(d_i\) equal to zero for all complexes \(K\) whose cohomology has no torsion, then for all such complexes \(K\) the Adams spectral sequence with term

\[ E_2=\operatorname{Ext}_{A^X}\bigl(X^*(K),X^*(P)\bigr) \]

exists and converges precisely to the stable homotopy groups \(\pi_*^s(K)\).

Corollary 1. For the spectrum \(X=MU\), \(X_{2n}=MU_n\), the Adams spectral sequence for any complex without torsion exists and converges to the stable homotopy groups \(\pi_*^s(K)\).

Remark. For the spectrum \(X=k\), \(X_0=BU\times Z\), \(X_{2n}=BU^{(2n)}\), where \(BU^{(2n)}\) is the \((2n-1)\)-connected space \(BU\), the conditions of Theorem 1 are not satisfied. The spectrum \(k\) is the spectrum of “stable” \(K\)-theory; here

\[ \Omega^{2n}X_{2n}=BU\times Z \]

(ordinary \(K\)-theory, where \(X_{2n}=BU\times Z\) and \(X_{2n-1}=U\), do not satisfy the requirement of “stabilization” \(\pi_{i-k}(X_i)=0\)).

II. The main question is the following: how are the Steenrod rings \(A^U\) and \(A^k\) to be computed for the theories of \(U\)-cobordisms and \(k\)-theory, respectively? We shall first compute the ring \(A^U\).

Consider the cobordism ring

\[ \Omega_U=U^*(P)=Z[x_1,\ldots,x_i,\ldots], \]

\(\dim x_i=-2i\) (see \((^{9-11})\)). The ring \(X^*(K)\), \(X=MU\), will always be denoted by \(U^*(K)\). Since \(\Omega_U=U^*(P)\), we have multiplication operations on the cohomology of a point \(x:\ U^j(K,L)\to U^{j-2k}(K,L)\), \(\dim x=-2k\), \(x\in\Omega_U^{2k}\), where \(\Omega_U^{2k}=U^{-2k}(P)\). Obviously, for any \(\alpha,\beta\in U^*(K,L)\) we have

\[ x(\alpha\beta)=(x\alpha)\beta=\alpha(x\beta). \]

We shall henceforth regard the ring \(A^U\) as a left \(\Omega_U\)-module. This module is free.

We note further that

\[ N=A^U\otimes_{\Omega_U}A^U \]

is also a left \(A^U\)-module; since \(N=U^*(MU\wedge MU)\). By virtue of the multiplication \(MU\wedge MU\to MU\), a “diagonal”

\[ \Delta:\ A^U\to A^U\otimes_{\Omega}A^U \]

is defined.

In \(U\)-theory there is a Künneth formula: there exists a natural homomorphism of \(A^U\)-modules

\[ U^*(K_1,L_1)\otimes_{\Omega_U} U^*(K_2,L_2) \to U^*(K_1\times K_2\,/\,K_1\times L_2\cup L_1\times K_2), \]

which is an isomorphism for complexes without torsion, and the algebra

\(A^U\) acts on \(U^*(K_1,L_1)\otimes_{\Omega_U}U^*(K_2,L_2)\) by means of the diagonal \(\Delta\) indicated above.

To construct operations from \(A^U\) we shall use an analogue of the Chern characteristic classes (see \((^7)\), §§ 1, 2).

Lemma 1. There exist unique Chern characteristic classes
\[ \sigma_k:k^0(K)\to U^{2k}(K), \]
having the following properties:

1. \(\sigma_0=1\); if \(\eta\) is a \(U_1\)-bundle, then
   \[ \sigma_1(\eta)\in \operatorname{Map}(K,MU_1). \]

2. The Whitney formula
   \[ \sigma(\xi\oplus\eta)=\sigma(\xi)\sigma(\eta),\qquad \sigma=\sum_i\sigma_i . \]

3. For any \(U_1\)-bundles \(\xi,\eta\), the class \(\sigma_1(\xi\oplus\eta)\) is represented in the form
   \[ \sigma_1(\xi)+\sigma_1(\eta)+a\bigl(\sigma_1(\xi)\sigma_1(\eta)\bigr)a\in A_2^U . \]

4. \[ \sigma_1(\lambda^{-1}\xi)=a_{n-1}\sigma_n(\xi),\qquad \xi\in \operatorname{Map}(K,BU_n),\quad a_{n-1}\in A^U . \]

As usual, the classes \(\sigma_k\) generate classes \(\sigma_\omega\), where \(\omega=(k_1,\ldots,k_s)\) is an unordered partition of the number \(k\) into positive summands \(k_i\) (or \(\omega=(0)\) and \(k=0\)), and \(\sigma_{(1,\ldots,1)}=\sigma_i\);
\[ \sigma_\omega(\xi\oplus\eta)= \sum_{\omega=(\omega_1,\omega_2)} \sigma_{\omega_1}(\xi)\sigma_{\omega_2}(\eta). \]

In \(U\)-theory there is a natural Thom isomorphism
\[ \varphi_U:U^*(K)\to U^*(M\xi,P), \]
where \(\xi\) is a complex bundle over \(K\), \(P\) is a point, and \(M\xi\) is the Thom complex. For manifolds \(M^n\) we shall denote by \(D\sigma_\omega(M^n)\) the dual (see \((^1)\)) bordism classes
\[ D\sigma_\omega\in U_{n-2k}(M^n) \]
(normal).

Lemma 2. For closed quasicomplex manifolds, the bordism classes
\[ D\sigma_\omega(M^n,\eta), \]
where \(\eta\) is the normal \(U\)-bundle to \(M^n\), determine unique homomorphisms
\[ \sigma_\omega^*:\Omega_U\to\Omega_U,\qquad \sigma_\omega^*\Omega_U^n\subset \Omega_U^{n-2k}, \]
\(\omega=(k_1,\ldots,k_s)\), \(\sum k_i=k\),
\[ \sigma_\omega^*[M^n]=\varepsilon D\sigma_\omega(M^n,\eta),\qquad [M^n]\in\Omega_U^n, \]
\[ \varepsilon:U_*(M^n)\to U_*(P) \]
is the natural homomorphism.

We note that for \(n=2k\), \(\sigma_\omega^*(M^n)\) are characteristic numbers; all the homomorphisms \(\sigma_\omega^*\) are easily computed for complex projective spaces:
\[ \sigma_\omega^*(CP^n)=\lambda_\omega[CP^{\,n-k}],\qquad \lambda_k=k. \]

From Lemma 1 follows the Leibniz formula
\[ \sigma_\omega^*(xy)= \sum_{(\omega_1,\omega_2)=\omega} \sigma_{\omega_1}^*(x)\sigma_{\omega_2}^*(y), \qquad xy\in\Omega_U . \]

With the help of Lemmas 1 and 2 one proves an important theorem which completely computes the ring
\[ A^U=\sum_i A_i^U . \]

Theorem 2. 1) There exist unique operations
\[ S_\omega:U^j(K,L)\to U^{j+2k}(K,L), \qquad \omega=(k_1,\ldots,k_s),\quad \sum k_i=k, \]
having the following properties:
a) the operations \(S_\omega\) commute with continuous mappings, the suspension isomorphism, and the homomorphism
\[ \delta:U^{j-1}(L)\to U^j(K,L); \]
b) for any \(\alpha,\beta\in U^*(K,L)\) the formula holds
\[ S_\omega(\alpha\beta)= \sum_{\omega=(\omega_1,\omega_2)} S_{\omega_1}(\alpha)S_{\omega_2}(\beta); \]
c) if
\[ \alpha\in \operatorname{Map}(K,MU_1)\subset U^2(K), \]
then
\[ r^0S_k(\alpha)=\alpha^{k+1},\qquad S_\omega(\alpha)=0,\quad \omega\ne(k); \]
d) the composition \(S_{\omega_1}\circ S_{\omega_2}\) is a linear combination, in the ring \(A^U\), of operations of the form \(S_\omega\);
e) if
\[ K=CP_1^N\times\cdots\times CP_n^N, \]
where \(n,N\) are large, and
\[ u_i\in U^2(CP_i^N) \]
are the elements dual to the submanifolds
\[ CP_i^{N-1}\subset CP_i^N, \]
\(u=u_1\cdots u_n\), then the operations \(S_\omega(u)\) are linearly independent for all \(\omega\), \(\dim\omega<n\), and the linear \(\mathbb Z\)-space spanned by \(S_\omega(u)\) is the ideal in the ring of symmetric polynomials in \(u_1,\ldots,u_n\) generated by the element \(u\);
f) the Chern classes \(\sigma_\omega\) are equal to
\[ \varphi_U^{-1}S_\omega\varphi_U(1). \]

2) Every element \(\gamma\in A_{2k}^U\) is represented uniquely in the form of a linear combination
\[ \sum_{i\to\infty}\lambda_i x_iS_{\omega_i}, \]
where \(x_i\) is some additive homogeneous basis of the ring \(\Omega_U\) and
\[ \dim(x_iS_\omega)=2k, \]
\(\lambda_i\) are integers.

The ring \(A^U\) is a graded topological ring with topological basis \(x_iS_\omega\), and the series
\[ \sum \lambda_i x_iS_{\omega_i} \]
converge if \(\dim\omega_i\to\infty\) as \(i\to\infty\).

The equality \(A^U=(\Omega_U\circ S)\) holds, where \(S\) is the ring generated by all \(S_\omega\).

3) The following commutation relation holds for the subrings \(\Omega_U\subset A^U\) and \(S\subset A^U\):

\[ S_\omega\circ x=\sum_{\omega=(\omega_1,\omega_2)}\sigma_{\omega_1}^{*}(x)\,S_{\omega_2}. \]

Adem’s formulas in the ring \(A^U\) follow completely from items c)—e) and 3) by the usual Cartan method \((^6)\).

Example. For \(K=S^0,\ L=P\) we have: the module \(M_U=U^*(P)\) is given by one generator \(t\in M_U^0\) and the relation \(S_\omega(t)=0,\ \dim\omega>0\).

Theorem 3. If there is a mapping \(A^U\xrightarrow{d}A^U\), where \(d(1)=S_\omega\), then the corresponding mapping
\[ d^*:\operatorname{Hom}_{A^U}(A^U,M_U)\to \operatorname{Hom}_{A^U}(A^U,M_U) \]
will coincide with \(\sigma_\omega^*:\Omega_U\to\Omega_U\), where \(\Omega_U\) is naturally isomorphic to the group \(\pi_*^S(MU)\), \(\pi_*^S(MU)=\operatorname{Hom}_{A^U}(A^U;M_U)\).

III. Let \(Q_p\) be the ring of rational numbers with denominator not divisible by \(p\). To study the \(p\)-component of the groups \(\pi_*^S(K)\) it suffices to study the ring \(A^U\otimes_{\mathbb Z}Q_p\) and the modules \(U^*(K)\otimes_{\mathbb Z}Q_p\). Let \(C\) be the class of finite groups of orders relatively prime to \(p\). As follows from \((^{9-11})\), the spectrum \(MU\) is \(C\)-homotopically equivalent to the direct sum \(\sum_\omega M_\omega\), where \(\omega\) are non-\(p\)-adic partitions \((k_1,\ldots,k_m)\), and the \(A\)-module \(H^*(M_\omega,\mathbb Z_p)\) is \(A/A\bar B\), \(B\) being generated by elements \(e_r'\in A^{2n-1}\) (recently Brown and Peterson in \((^5)\) constructed the spectrum \(M_\omega\)). Let \(X=M_\omega\) and \(A_p^U=X^*(X)\otimes_{\mathbb Z}Q_p\) be a graded ring over \(Q_p\). The following simple theorem holds.

Theorem 4. A. The ring \(A^U\otimes_{\mathbb Z}Q_p\) is isomorphic to \(GL(A_p^U)\), where \(GL(A_p^U)\) is the graded ring of matrices over \(A_p^U\) of the form \((a_{\omega_i,\omega_j})\in GL(A_p^U)\), \(a_{\omega_i,\omega_j}\in A_p^U\),

\[ n=\dim\omega_i-\dim\omega_j+\dim a_{\omega_i,\omega_j} \]

is the dimension of the matrix.

B. In the ring \(A_p^U\) lies the subring
\[ \Omega_U(p)=\mathbb Z[x_1,\ldots,x_i,\ldots]\subset \Omega_U\otimes Q_p,\qquad \dim x_i=2p^i-2, \]
a projector
\[ \pi_p:\Omega_U\otimes Q_p\to\Omega_U(p) \]
is defined,
\[ \pi_p(xy)=\pi_p(x)\pi_p(y); \]
the generators \(x_i\) are such that
\[ \sigma_{(2p^i-2)}^{*}x_i=p \]
and, for all \(\omega\), \(\dim\omega=2p^i-2\),
\[ \sigma_\omega^*x_i=0 \pmod p \]
(the remaining choice of the \(x_i\) is arbitrary, but they are fixed).

C. The algebra \(S_p\) consists of all elements \(\alpha\) of the algebra \(S\otimes Q_p\), and they are identified if
\[ \pi_p\alpha_1^*\pi_p=\pi_p\alpha_2^*\pi_p \]
on \(\Omega_U(p)\), where \(\alpha^*:\Omega_U\to\Omega_U\) are the homomorphisms of Lemma 2.

D. \(X^*(P)=M_U(p)\), where \(M_U(p)\) has a generator \(f\) and is given by the relation
\[ S_p(t)=0 \]
(\(M_U, M_U(p)\) are the corresponding modules for the sphere).

Although the description of the ring \(A_p^U\) in Theorem 4 is algebraically complete, it is inconvenient for computations, and it would be better to find direct composition formulas.

IV. We now turn to \(k\)-theory, where \(k_{2n}=BU^{(2n)}\) and \(k_0=BU\times Z,\ k_2=BU,\ k_4=BSU,\ \Omega^{2n}k_{2n}=BU\times Z\) (Bott), and embeddings \(x:k_{2n}\to k_{2n-2}\) are defined for all \(n\). For \(k\)-theory so defined we have that the functors \(k^i(K,L)\) are isomorphic to the ordinary ones for \(i\le 0\), and for \(i>0\), \(k^{2i}(K,L)\) consists of all elements of \(K^0\) of filtration \(\ge 2i\) for complexes without torsion. There is a Bott operator
\[ x:k^j(K,L)\to k^{j-2}(K,L), \]
representing an element of dimension \((-2)\) from the Steenrod ring \(A^k\). We use two approaches to computing the ring \(A^k\): the second of them uses Adams operations
\[ \Psi^k:K^0\to K^0 \]
and the Bott operator
\[ x:k^j\to k^{j-2}, \]
while the first is based on the immersion \(k^2\to U^2\) \((^7)\). Both are incomplete.

1st method: we use the “corrected” Conner–Floyd operators
\[ \lambda_{-1}:U^j\to k^j,\quad |j|<\infty \]
and
\[ \sigma_1:k^2\to U^2, \]
which give a splitting of theories

cohomology \(U^2=k^2+\cdots,\ \sigma_1\circ\lambda_{-1}: U^2\to U^2\) is the projector (7); we have an embedding \(x^N A_j^k\to A_{j-2N}^U,\ N\) large. Note that \(x^N\to a_N\) (see Lemma 1) and \(\sigma_1\lambda_{-1}(\xi)=a_{N-1}\sigma_N(\xi)\), where \(\xi\) is a \(U_N\)-bundle, \(x\) is the Bott operator.

The 2nd method of defining operations in \(k\)-theory: the Adams operations \(\Psi^k\) (see (3)) do not exist in stable \(k\)-theory, since \(\Psi^k\circ x=kx\circ\Psi^k\), but there do exist unstable operations \(k^n\Psi^k:k^{2n}(K,L)\to k^{2n}(K,L)\). Let \(n\) be large. We choose a large number \(m\) and form a linear combination \(\sum_k \lambda_k^{(n)} k^n\Psi^k=a_n\) such that the mappings of homotopy groups
\(a_{n*}^{(j)}:\pi_{2n+2j}(BU^{(2n)})\to\pi_{2n+2j}(BU^{2n})\), for \(j\le m\), do not depend on \(n\) in the sense that for all \(N>n\) one can find numbers \(\lambda_k^{(N)}\) such that the homomorphisms

\[ a_{n*}^{(j)}=\sum_k \lambda^{(n)} k^{\,n+j} \]

for \(j\le m\) coincide with

\[ a_{N*}^{(j)}=\sum_k \lambda_k^{(N)} k^{\,N+j}. \]

Such a sequence \(a=(a_n)\), where \(a_{n*}^{(j)}\) does not depend on \(n\) for \(j\le m(n)\to\infty\), we shall call an operation in \(k\)-theory. If \(a_{n*}^{(j)}=0\) for \(j<q\), then \(a=x^q b\), where \(x\) is the Bott operator and \(b:k^l\to k^{l+2q}\). We denote the ring of operations so constructed (together with \(x\)) by \(A_\Psi^k\). If one takes the sphere module \(M_k^\Psi\) with one generator \(t\) over \(A_\Psi^k\), and such that \(bt=0\) for all positive dimensions, then the following fact holds: \(\operatorname{Ext}_{A_\Psi^k}^{1,2i}(M_k,M_k)\) is a cyclic group of order \(d_i\), where \(d_i\) is the greatest common divisor of all the numbers \(k^n(k^i-1)\), over all \(k\) and for large \(n\).

V. We indicate here some of the simplest results of computations.

Theorem 5. 1) The group \(\operatorname{Ext}_{A_\Psi^k}^{1,2i}(M_k,M_k)\) is \(\mathbb Z_{d_i}\), where \(d_i\) is the greatest common divisor of the numbers \(k^n(k^i-1)\) over all \(k\), \(n\to\infty\).

2) The group \(\operatorname{Ext}_{AU}^{1,4i+2}(M_U,M_U)\) is \(\mathbb Z_2\).

3) The groups \(\operatorname{Ext}_{AU}^{1,4i}(M_U,M_U)\) are \(\mathbb Z_{d_i/a_i}\), where \(a_{2q}=1,\ a_{2q+1}=2,\ d_i/2\) is the denominator of the fraction \(B_i/2i,\ B_i\) is a Bernoulli number.

4) \(d_i/\operatorname{Ext}_{AU}^{1,2i}(M_U,M_U)=0,\ i\ne 4k-1;\ d_3/E_3^{1,8k+6}\ne 0\), where \(d_i\) are the differentials of the Adams spectral sequence \((E_r,d_r)\), \(k\ge0\).

5) The homomorphism \(q=\operatorname{Ext}^1\circ J:\pi_{2i-1}(SO)\to\pi_{N+2i-1}(S)^N\to\operatorname{Ext}_{AU}^{1,2i}\) coincides exactly with the Hopf–Milnor–Kervaire invariant (8).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
17 VIII 1966

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