Reports of the Academy of Sciences of the USSR

1. Volume 133, No. 5

MATHEMATICS

A. M. VINOGRADOV

ON THE ADAMS SPECTRAL SEQUENCE

(Presented by Academician P. S. Aleksandrov on 13 V 1960)

The purpose of this note is to investigate the structure of the limiting term of the spectral sequence that was introduced by Adams in \((^1)\). In what follows this spectral sequence will simply be called the Adams sequence. Let us note at once that Adams’ assertion on the structure of the term \(E_\infty(S^0)\) (\(S^0\) is the zero-dimensional sphere) will follow directly from our general result.

Let \(X\) be a space and let \(S\) be the suspension operator. We put
\[ \pi_m^S(X)=\operatorname{Dir}\lim_{n\to\infty}[\pi_{m+n}(S^nX)]; \qquad \pi_*^S(X)=\sum_{m\ge 0}\pi_m^S(X); \]
\(K_m^S(X)\) is the subgroup of the group \(\pi_m^S(X)\) consisting of elements of order \(q\ne p\), where \(p\) is a fixed prime; \(K_m(X)\) is the subgroup of the group \(\pi_m(X)\) consisting of elements of order \(q\ne p\);
\[ K_*^S(X)=\sum_{m\ge 0}K_m^S(X); \qquad K_*(X)=\sum_{m\ge 0}K_m(X). \]

We shall denote the Steenrod algebra modulo \(p\), graded according to the degree of operations, by
\[ A=\sum_{q\ge 0}A_q, \]
and all groups of the form \(H^*(X; Z_p)\) will be regarded as \(A\)-modules. The field \(Z_p\) can also be made into an \(A\)-module if one sets \(A_q\cdot Z_p=0,\ q>0\), and, since \(A_0\cong Z_p,\ A_0\cdot Z_p=Z_p\cdot Z_p\). Therefore one can define the groups
\[ \operatorname{Ext}_A^{s,t}(H^*(x; Z_p); Z_p). \]
We can now formulate the main theorem of \((^1)\).

Theorem 1 (Adams). Let \(X\) be a space such that the groups \(H^t(X; Z_p)\) have finite type for all \(t\). Then there exists a spectral sequence \(\{E_r^{s,t}(X), d_r\}\) satisfying the following conditions:

1) \(E_r^{s,t}(X)=0\), if \(s<0\) or \(t<s\);

2) \(d_r:E_r^{s,t}(X)\to E_r^{s+r,t+r-1}\);

3) \(E_2^{s,t}(X)\cong \operatorname{Ext}_A^{s,t}(H^*(X; Z_p), Z_p)\);

4) there exists a canonical monomorphism \(E_R^{s,t}\subset E_r^{s,t}\) for \(s<r\le R<\infty\),
\[ E_\infty^{s,t}=\bigcap_{s<r<\infty}E_r^{s,t}; \]

5) there exist such subgroups \(B^{s,t}\) of the group \(\pi_{t-s}^S(X)\) that
\[ \cdots\subset B^{s,t}\subset B^{s-1,t-1}\subset\cdots\subset B^{0,t-s}=\pi_{t-s}^S(X), \qquad E_\infty^{s,t}=B^{s,t}/B^{s+1,t+1}; \]

6)
\[ \bigcap_{t-s=m}B^{s,t}=K_m^S(X). \]

This theorem contains almost no information about the structure of the groups \(B^{s,t}\). For the case \(X=S^0\), Adams proved the assertion that these groups are arranged as follows. Let \(\alpha\in\pi_{m+n}(S^n)\) \((m<n-1)\) define an element \(\alpha'\in\pi_m^S(S^0)\). Then \(\alpha'\) has filtration \(s\), i.e. \(\alpha'\in B^{s,t}\) and \(\alpha'\notin B^{s+1,t+1}\), where \(m=t-s\), if and only if in the Adams complex

\(K_\alpha = e^{m+n+1}\underset{\alpha}{\cup} S^n\) there acts nontrivially a cohomology operation

\[ Q:H^n(K_\alpha;\; Z_p)\to H^{n+m+1}(K_\alpha;\; Z_p) \]

of order \(s\), and every cohomology operation of order \(<s\) acting according to the same scheme is trivial in \(K_\alpha\).

We now give the general results that we shall need.

Lemma 1. For any space \(X\) and prime \(p\) there exists a space \(X_p\) and a map \(f^p:X\to X_p\) such that \(\operatorname{coker} f^p_* = 0\), \(\ker f^p_* = K_*(\dot X)\). The space \(X_p\), possessing the indicated properties, is determined uniquely up to singular homotopy type (we say that spaces have the same singular homotopy type if their natural systems are isomorphic).

It follows from this lemma that the homotopy groups mod \(p\) of the spaces \(X\) and \(X_p\) are arranged in the same way.

Definition 1. A space \(X\) is called a \(p\)-space if it has the same singular homotopy type as \(X_p\).

From the construction of the Adams sequence it follows:

Lemma 2. The Adams sequences mod \(p\), computed for the spaces \(X\) and \(X_p\), are isomorphic, and this isomorphism is realized by the map \(f^p_*\).

This lemma shows that, for the study of the Adams sequence, it is enough to regard \(X\) as a \(p\)-space.

Definition 2. A \(p\)-system of some \(p\)-space \(X\) is a system of fibrations

\[ X_0 \xleftarrow[\rho_1]{F_1} X_1 \xleftarrow[\rho_2]{F_2} X_2 \xleftarrow[\rho_3]{F_3}\cdots, \tag{1} \]

satisfying the conditions:

1) the limiting space \(\bar X\) of this system of fibrations has the same singular homotopy type as the space \(X\);

2) the spaces \(X_0, F_1, F_2,\ldots\) are direct products of spaces of type \(K(Z_p,n)\) (the numbers \(n\) in one and the same product may be different);

3) the natural projections \(p_j:\bar X\to X_j\) induce epimorphisms \((p_j)_*\) for all \(j\ge 0\).

In the class of all \(p\)-systems of the space \(X\) one can introduce a relation of partial order. Namely, let \(\alpha=\{X_i,\rho_i,F_i,p_i\}\) and \(\beta=\{X'_i,\rho'_i,F'_i,p'_i\}\) be two \(p\)-systems. Then \(\alpha>\beta\) if, identifying the groups \(\pi_*(X)\), \(\pi_*(\bar X)\), and \(\pi_*(\bar X')\), \(\ker(p_i)_*=\ker(p'_i)_*\), \(i<n\), and \(\ker(p_n)_*\subset \ker(p'_n)_*\). We can now formulate the following main theorem.

Theorem 2. In the class of \(p\)-systems of a certain space \(X\) there is one and only one \(p\)-system (up to singular homotopy type) which is maximal with respect to the order introduced.

Theorem 2 allows us to define the notion of a natural filtration that we need.

Definition 3. The natural filtration of a \(p\)-space is the sequence of subspaces of the space \(\bar X\):

\[ \bar X=Y_0\supset Y_1\supset\cdots\supset Y_s\supset\cdots, \]

where \(Y_i=p_i^{-1}(F_i)\), \(\{X_i,\rho_i,F_i,p_i\}\) is the maximal \(p\)-system of the space \(X\), and \(\bar X\) is its limiting space.

Definition 4. We shall call a filtration of the homotopy and homology groups of a \(p\)-space \(X\) natural if it is generated by the natural filtration of the space \(X\).

We note that the inclusions \(i_k:Y_k\subset \widetilde X\) induce monomorphisms \((i_k)_*\), i.e. the groups \(\pi_*(Y_k)\) “realize” precisely all elements of filtration \(\geq k\).

Digressing somewhat, we observe that the notion of a maximal \(p\)-system of the space \(X\) determines a method for computing the homotopy groups \(\pi_i(X)\) by “killing cohomology operations” of various orders, in contrast with the classical method of “killing homotopy groups.” This follows from the fact that the space \(X_s\) is a universal example for constructing certain operations of order \(s+1\).

The following lemma makes it possible to determine the natural filtration in the groups \(\pi_m^s(X)\).

Lemma 3. The elements \(\alpha\in \pi_{m+n}(S^nX)\) and \(E\alpha\in \pi_{m+n+1}(S^nX)\), where \(E\) is the suspension, have the same natural filtration if \(m<n-1\).

Theorem 3. The filtration of the group \(\pi_*^s(X)\) defined by the Adams sequence coincides with the natural filtration of this group.

We shall show how Theorem 3 can be proved on the basis of Theorem 2. Suppose that \(\pi_i(X)=0\) for \(i<n\), where \(n\) is very large. In the contrary case one may consider the space \(S^nX\). Let, further, \(\{X_i,\rho_i,F_i,p_i\}\) be a maximal \(p\)-system of the space \(X\) and let the sequence

\[ 0\leftarrow H^*(X;Z_p)\xleftarrow{\varepsilon} C_0=\sum_{t\geq 0} C_{0,t}\xleftarrow{d} C_1=\sum_{t\geq 0} C_{1,t}\xleftarrow{d}\cdots \]

be a free acyclic \(A\)-resolution of the \(A\)-module \(H^*(X;Z_p)\). Since the \(A\)-module \(C_0\) is free and, as we suppose, is of finite type, it can be realized up to dimension \(2n-1\) as the cohomology module of some direct product \(B\) of Eilenberg–Mac Lane spaces. Therefore there exists a map, unique up to homotopy,
\(f:X\to B\), satisfying the condition \(f^*=\varepsilon\) in dimensions \(<2n-1\), if one takes into account the identification of \(C_0\) and \(H^*(B;Z_p)\). We now replace the map \(f\) by the fibration \(f:\widetilde X\to B\) and suppose that the fibre of this fibration is the space \(\Phi\). Then, as follows from the construction of the Adams sequence, the inclusion \(\mu:\Phi\subset \widetilde X\) determines all elements of filtration \(1\) in the sense of Adams, i.e. these elements form the subgroup \(\operatorname{im}\mu_*\).

Since \(\operatorname{coker}\varepsilon=0\), the space \(B\) can be constructed in the following way. Choose in the \(A\)-module \(H^*(X;Z_p)\) a minimal homogeneous system of generators \(u=\{u_i\}\), \(i=1,2,\ldots,N\), and with each element \(u_i\) associate the space \(K_i=K(Z_p;\deg u_i)\). Putting

\[ B=\sum_{i=1}^{N}K_i, \]

we obtain the required representation. The map \(\varepsilon=f^*\) can then be specified by the relations \(\varepsilon(\overline v_i)=u_i\), where \(\overline v_i\) is the image in \(H^*(B;Z_p)\) of the fundamental class \(v_i\) of the space \(K_i\) under the projection \(B\to K_i\). We note at once that, if

\[ X_0=\sum_{i=1}^{l} K(Z_p;m_i), \]

then, according to the definition of a \(p\)-system, the images under \(p_0^*\) of the fundamental classes of the spaces \(K(Z_p;m_i)\) (or their multiples modulo \(p\)) enter into the system \(u\). Therefore, for \(i\leq l\), we shall identify \(K_i\) and \(K(Z_p;m_i)\).

We now realize the fibration \(\widetilde f\). For this purpose consider the fibration

\[ \tau:\widetilde X\to \widetilde X\times \sum_{i>l}K_i=T, \]

which is the Whitney sum of fibrations whose Eilenberg–Mac Lane invariants are equal to \((u_i'-v_i')\), where \(u_i'\), \(v_i'\) are the images of the classes \(u_i\), \(v_i\), respectively, in \(H^*(T;Z_p)\) under the corresponding

projections. The space \(\overline{\overline X}\) has the same singular homotopy type as \(X\). Indeed, \(\overline{\overline X}\) is the Whitney sum of fibrations
\[ \lambda_i:\overline{\overline X}_i \xrightarrow{E_\alpha} \overline X, \]
whose fibers \(E_\alpha\) are contractible, if one sets \(\lambda_i=r_i\tau_i\), where the fibration
\[ \tau_i:\overline{\overline X}_i \to \overline X \times K_i \]
is determined by the Eilenberg—MacLane invariant equal to
\[ (u_i\otimes 1-1\otimes v_i)\in H^*(\overline X\times K_i;Z_p), \]
and \(r_i:\overline X\times K_i\to \overline X\) is the projection. The equivalence of the spaces \(\overline{\overline X}\) and \(\overline X\) is generated, obviously, by the mapping \(r\circ\tau\), where \(r:T\to\overline X\) is the projection. On the other hand, the fibration
\[ p_0\times 1:T\to\overline X\times\left(\sum_{i>l}A_i\right)\to X_0\times\left(\sum_{i>l}K_i\right)=B. \]

It turns out that
\[ f'=(p_0\times 1)\circ\tau:\overline X\to B \]
and there is a realization of the fibration \(\widetilde f\), since for \(i>l\)
\[ f^*(v_i)=\tau^*(p_0\times 1)^*(v_i)=\tau^*(v_i)=\tau^*(u_i)=(r\circ\tau)^*(u_i), \]
and, similarly,
\[ f^*(v_i)=(r\circ\tau)^*(u_i) \]
for \(i<l\). The fiber \(\Phi'\) of the fibration \(f'\), as follows from the construction of \(f'\), can be represented in the form of the Whitney sum of fibrations
\[ \nu_i:M_i\to \overline Y_1,\quad i>l, \]
where
\[ \overline Y_1=(r\circ\tau)^{-1}(Y_1)\bigl(Y_1=p_1^{-1}(F_1)\bigr), \]
and \(\nu_i\) is determined by the Eilenberg—MacLane invariant
\[ i^*[(r\circ\tau)^*(u_i)], \]
where
\[ \overline Y_i\to\overline X \]
is an inclusion. It follows from this that
\[ \operatorname{im}\mu_*=\operatorname{im}i_*, \]
where \(\mu':\Phi'\subset\overline X\) is an inclusion, which shows that the elements of filtration \(\ge l\) in the sense of Adams have natural filtration \(\ge l\), and conversely.

Noting now that the elements of the group \(\pi_*(\overline Y_1)\), whose natural filtration is \(\ge1\), coincide under \(i_*\) with the elements of the group \(\pi_*(X)\) whose natural filtration is \(\ge2\), and similarly for the Adams filtration, we can continue our argument by induction. Theorem 3 is proved.

Adams’ assertion. We shall now derive Adams’ assertion from Theorem 3. Let
\[ \alpha'\in\pi_m^S(S^0) \]
have filtration \(s\) in the sense defined above. Then one may suppose that \(\alpha'\) is determined by a generating element \(\alpha\) of some space of type
\[ K(Z_p;n+m)\in F_i, \]
where \(\{X_i,\rho_i,F_i,p_i\}\) is a maximal \(p\)-system of the sphere \(S^n\). This means that \(\alpha\) “kills” a certain element of
\[ H^{m+1}(X_{i-1}), \]
i.e. an example of an operation of order \(\le i\). By virtue of the maximality of the \(p\)-system, the order of this operation \(P\) is exactly \(i\). Attaching the element \(\alpha\) leads to the operation \(P\) ceasing to be “killed,” and hence, in the complex
\[ e^{m+n+1}\cup_\alpha S^n \]
it will be nontrivial. Operations of order \(<i\) will still remain trivial, i.e. \(s=i\).

Moscow State University
named after M. V. Lomonosov

Received
11 V 1960

REFERENCES

1. J. F. Adams, Comm. Math. Helv., 32, No. 3, 180 (1958).