# Cohomology operation

A natural transformation of certain cohomology functors into others (most often — into themselves). By a cohomology operation of type $( n , m ; \pi , G )$, $n$ and $m$ being integers and $\pi , G$ Abelian groups, one means a family of mappings (not necessarily homomorphisms) between cohomology groups $\theta _ {X} : H ^ {n} ( X ; \pi ) \rightarrow H ^ {m} ( X ; G )$, defined for any space $X$, such that $\theta _ {X} \circ f ^ {*} = f ^ {*} \circ \theta _ {Y}$ for any continuous mapping $f : X \rightarrow Y$( naturality). The set of all cohomology operations of type $( n , m ; \pi , G )$ forms an Abelian group with respect to the composition: $( \theta + \psi ) _ {X} = \theta _ {X} + \psi _ {X}$, and is denoted by $O ( n , m ; \pi , G )$.

Examples of cohomology operations. The Steenrod reduced powers $Sq ^ {i}$ and ${\mathcal P} ^ {i}$( cf. Steenrod reduced power); the Pontryagin square ${\mathcal P} _ {1}$; the Postnikov square; raising to the $k$- th power, $\mu _ {k}$: for $x \in H ^ {s} ( X ; \pi )$, where $\pi$ is a ring, $\mu _ {k} ( x) = x ^ {k}$, $\mu _ {k} \in O ( s , ks ; \pi , \pi )$; the Bockstein homomorphism $\beta$; cohomology operations induced by homomorphisms $\pi \rightarrow G$ of the coefficient groups, for example, $\mathop{\rm mod} p : H ^ {n} ( X) \rightarrow H ^ {n} ( X , \mathbf Z _ {p} )$.

Cohomology operations represent an additional structure in cohomology functors, and for this reason they make it possible to solve problems of homotopic topology that are not solvable "at the level" of cohomology groups. Examples. 1) Let $X$ and $Y$ be two spaces and let $x \in H ^ {n} ( X ; \pi )$, $y \in H ^ {n} ( Y ; \pi )$ be two elements. Does there exist a mapping $f : X \rightarrow Y$ such that $f ^ {*} ( y) = x$? A first sufficient condition for the absence of such an $f$ is the absence of a homomorphism $g : H ^ {*} ( Y , \pi ) \rightarrow H ^ {*} ( X , \pi )$ with $g ( y) = x$. (By this method, one can prove, for example, the Brouwer theorem on fixed points.) If $g$ exists, then the non-existence of $f$ can be established as follows: Let $\theta$ be a cohomology operation, $\theta \in O ( n , m ; \pi , G )$, with $\theta _ {Y} ( y) = 0$, $\theta _ {X} ( x) \neq 0$. Then $0 \neq \theta _ {X} ( x) = \theta _ {X} ( f ^ {*} ( y) ) = f ^ {*} ( \theta _ {Y} ( y)) = f ^ {*} ( 0) = 0$, which is impossible. 2) Is the mapping: $f : S ^ {m} \rightarrow S ^ {n}$ essential? Let $C _ {f} = S ^ {n} \cup f e ^ {m}$. Then (for $m \neq n$) $H ^ {m+} 1 ( C _ {f} ; G ) = G$, $H ^ {n} ( C _ {f} ; \pi ) = \pi$. If there is a cohomology operation $\theta \in O ( n , m+ 1 ; \pi , G )$ with $\theta _ {C _ {f} } \neq 0$, then $f$ is essential. In this case, the operation $\theta$ detects the mapping $f$ or the element $[ f ] \in \pi _ {m} ( S ^ {n} )$.

There is an isomorphism of groups $O ( n , m ; \pi , G ) \approx H ^ {m} ( K ( \pi , n ) ; G )$, where $K ( \pi , n )$ is an Eilenberg–MacLane space, and therefore $O ( n , m ; \pi , G ) \approx [ K ( \pi , n ) , K ( G , m ) ]$( see Representable functor). The groups $H ^ {m} ( K ( \pi , n ) ; G )$ have been computed for all $m$ and $n$ and any finitely generated $\pi$ and $G$[9].

The cohomology suspension ${} ^ {1} \theta \in O ( n- 1 , m- 1 ; \pi , G )$ of a cohomology operation $\theta \in O ( n , m ; \pi , G )$ is the mapping ${} ^ {1} \theta _ {X}$ given by the composition

$$H ^ {n-} 1 ( X ; \pi ) \rightarrow H ^ {n} ( SX ; \pi ) \rightarrow ^ { {\theta _ SX} } \ H ^ {m} ( SX ; G ) \rightarrow H ^ {m-} 1 ( X ; G ) ,$$

where $SX$ is the suspension of $X$. For example, ${} ^ {1} \mu _ {k} = 0$, ${} ^ {1} Sq ^ {i} = Sq ^ {i}$, ${} ^ {1} {\mathcal P} ^ {i} = {\mathcal P} ^ {i}$. When $m \leq 2n- 1$, $\theta \rightarrow {} ^ {1} \theta$ is an isomorphism. For any $X$, ${} ^ {1} \theta _ {X}$ is a group homomorphism.

By a stable cohomology operation of type $( \pi , G )$ and of degree $k$ one means a set $\{ \theta _ {n} \} _ {- \infty } ^ {+ \infty }$ with $\theta _ {n} \in O ( n , n+ k ; \pi , G )$ and ${} ^ {1} \theta _ {n} = \theta _ {n-} 1$. Such cohomology operations form an Abelian group $O _ {S} ( k ; \pi , G )$, isomorphic to the group $\lim\limits _ {\leftarrow n } H ^ {n+} k ( K ( \pi , n ) ; G )$, the latter being the inverse limit of the sequence

$$\dots \leftarrow H ^ {n+} k ( K ( \pi , n ) ; G ) \leftarrow$$

$$\leftarrow \ H ^ {n+} k- 1 ( K ( \pi , n- 1 ) ; G ) \leftarrow \dots .$$

The group $\oplus _ {k} O _ {S} ( k ; \pi , G )$ is denoted by $O _ {S} ( \pi , G )$.

Examples of stable cohomology operations. The Steenrod powers $Sq ^ {i} \in O _ {S} ( i ; \mathbf Z _ {2} , \mathbf Z _ {2} )$ and ${\mathcal P} ^ {i} \in O _ {S} ( 2 ( p- 1 ) i ; \mathbf Z _ {p} , \mathbf Z _ {p} )$( where $p > 2$ is a prime number), and the Bockstein homomorphism $\beta \in O _ {S} ( 1 ; \mathbf Z _ {m} , \mathbf Z _ {m} )$.

If $\theta \in O ( n , m ; \pi , G )$ and $\phi \in O ( m , l ; G , \tau )$, then the cohomology operation $\phi \circ \theta \in O ( n , l ; \pi , \tau )$ is defined. In particular, one can define the composite $\phi \circ \theta \in O _ {S} ( \pi , \tau )$ of any two stable cohomology operations $\theta \in O _ {S} ( \pi , G )$ and $\phi \in O _ {S} ( G , \tau )$, so that the group $O _ {S} ( \pi , \pi )$ is a ring; $O _ {S} ( \mathbf Z _ {p} , \mathbf Z _ {p} )$ is called the Steenrod algebra $A _ {p}$.

Cohomology operations first emerged in the solution of the problem of the classification of mappings of an $( n+ 1 )$- dimensional polyhedron into an $n$- dimensional sphere ( $n = 2$ in [1] and $n > 2$ in [2]). The classification theorem [2]: There is an exact sequence of groups:

$$0 \rightarrow \ \mathop{\rm Coker} ( Sq ^ {2} : H ^ {n-} 1 ( X ; \mathbf Z _ {2} ) \rightarrow H ^ {n+} 1 ( X ; \mathbf Z _ {2} ) ) \rightarrow$$

$$\rightarrow \ [ X , S ^ {n} ] \rightarrow \mathop{\rm ker} \left ( H ^ {n} ( X) \mathop \rightarrow \limits ^ { { \mathop{\rm mod}} 2 } H ^ {n} ( X ; \mathbf Z _ {2} ) \mathop \rightarrow \limits ^ { {Sq ^ {2} }} H ^ {n+} 2 ( X ; \mathbf Z _ {2} ) \right ) \rightarrow 0 .$$

The extension theorem [2]: Let $Y$ be an $( n+ 2 )$- dimensional polyhedron and let $Y ^ {n+} 1$ be its $( n+ 1 )$- dimensional skeleton. A mapping $f : Y ^ {n+} 1 \rightarrow S ^ {n}$ defines an element $y = f ^ {*} ( s) \in H ^ {n} ( Y ^ {n+} 1 )$, where $s \in H ^ {n} ( S ^ {n} )$ is a generator. This mapping can be extended to $g : Y \rightarrow S ^ {n}$ if and only if $Sq ^ {2} ( z \mathop{\rm mod} 2 ) = 0$, where $j ^ {*} z = y$ for the inclusion $j : Y ^ {n+} 1 \rightarrow Y$.

Corresponding to the cohomology operation $\theta \in O ( n , m ; \pi , G )$, the mapping $\theta : K ( \pi , n ) \rightarrow K ( G , m )$ induces from the standard Serre fibration

$$K ( G , m- 1 ) \rightarrow \ PK ( G , m ) \rightarrow \ K ( G , m )$$

the fibration

$$F = K ( G , m- 1 ) \rightarrow ^ { i } \ E ( \theta ) \rightarrow ^ { p } \ K ( \pi , n ) = B .$$

Secondary cohomology operations are cohomology classes of spaces $E ( \theta )$. More precisely, let $\phi \in H ^ {q} ( E ( \theta ) ; H )$ be given, where $H$ is an Abelian group. There exists for any $u \in H ^ {n} ( X , \pi )$ with $\theta _ {X} ( u) = 0$ an element $\widetilde{u} \in [ X , E ( \theta ) ]$ with $p _ {\#} \widetilde{u} = u$, where $p _ {\#} : [ X , E ( \theta ) ] \rightarrow [ X , B ]$ is the mapping induced by $p$; the element $v = \phi \circ \widetilde{u} = \widetilde{u} {} ^ {*} ( \phi ) \in H ^ {q} ( X ; H )$ depends on the choice of the element $\widetilde{u}$. The arbitrariness in the choice of $\widetilde{u}$ is determined by the inverse image $p ^ {-} 1 ( u)$, that is, by the orbit of the action of the group $[ X , F ] = H ^ {m-} 1 ( X ; G )$ on the set $[ X , E ( \theta ) ]$. When $m < 2n- 1$ and $q < 2n- 1$, $i _ {\#}$ and $\phi _ {\#}$ are group homomorphisms, and, therefore, under a different choice of $\widetilde{u}$ the element $v$ can change only by some element of the subgroup $\mathop{\rm Im} ( i ^ {*} \phi : H ^ {m-} 1 ( X ; G ) \rightarrow H ^ {q} ( X ; H ))= Q$ of the group $H ^ {q} ( X , H )$. One defines the secondary cohomology operation $\Phi$ by setting $\Phi _ {X} ( u) = u + Q$( the coset $u + Q$ is uniquely determined by the element $u$). Thus, the mapping $\Phi _ {X}$ is defined on the subgroup $\mathop{\rm Ker} \theta _ {X}$ and takes values in the quotient space $H ^ {q} ( X ; H ) / Q$, where $Q$ is called the indeterminacy of the cohomology operation $\Phi$. An alternative terminology is that $\Phi$ is a partial multi-valued cohomology operation from $H ^ {n} ( \cdot ; \pi )$ into $H ^ {q} ( \cdot ; H )$.

Secondary cohomology operations are natural in the following sense: For any $f : Y \rightarrow X$ and any $u \in \mathop{\rm Ker} \theta \subset H ^ {n} ( X ; \pi )$, one has $f ^ {*} ( \Phi _ {X} ( u) ) \subset \Phi _ {Y} ( f ^ {*} ( u) )$[3]. If $i ^ {*} ( \phi ) = 0$, then $\phi = p ^ {*} ( \theta ^ \prime )$ for some $\theta ^ \prime \in H ^ {q} ( B ; H )$, so that $\widetilde{u} {} ^ {*} ( \phi ) = \theta ^ \prime ( u)$, and therefore $\Phi u = u ^ {*} ( \theta ^ \prime ) = \theta ^ \prime ( u)$ with zero indeterminacy. Here, $\theta ^ \prime ( u) = \theta ^ {\prime\prime} ( u)$, where $\theta ^ {\prime\prime}$ is any cohomology operation in $H ^ {m} ( B ; H )$ such that $p ^ {*} ( \theta ^ {\prime\prime} ) = \phi$, and $u \in \mathop{\rm Ker} \theta$, so that the cohomology operation $\Phi$ is a uni-valued cohomology operation $\theta ^ \prime$ restricted to $\mathop{\rm Ker} \theta$.

To each secondary cohomology operation there corresponds a relation among the ordinary (primary) cohomology operations. If $q < 2m - 1$, then the cohomology operation $i ^ {*} ( \phi ) \in H ^ {q} ( K ( G , m- 1 ) ; H )$ is uniquely representable in the form ${} ^ {1} \psi$ with $\psi \in H ^ {q+} 1 ( K ( G , m+ 1 ) ; H )$ and $\psi \circ \theta = 0$. If $\phi ^ \prime \in H ^ {m} ( E ( \theta ) ; H )$ is such that $i ^ {*} ( \phi - \phi ^ \prime ) = 0$, then corresponding to the cohomology operation $\phi ^ \prime$ is the same relation $\psi \circ \theta = 0$. Conversely, there corresponds to any relation of the form $\psi \circ \theta = 0$ a set of secondary cohomology operations $\{ \Phi \}$, any two of which differ from each other by a primary cohomology operation defined on the kernel of $\theta$.

A more general notion of a secondary cohomology operation is obtained by starting from a set $\theta = ( \theta _ {1} \dots \theta _ {n} )$ with $\theta _ {i} \in H ^ {m _ {i} } ( K ( \pi , n ) ; G )$ and the relation $\sum \psi _ {i} \circ \theta _ {i} = 0$( see [3]).

Example of a secondary cohomology operation. Let

$$\theta = S q ^ {2} \circ \mathop{\rm mod} 2 : \ H ^ {n} ( \cdot ; \mathbf Z ) \ \rightarrow H ^ {n+} 2 ( \cdot ; \mathbf Z _ {2} )$$

and let $\phi \in H ^ {n+} 3 ( E ( \theta ) ; \mathbf Z _ {2} )$ be a $\mathbf Z _ {2}$- generator. This gives rise to a secondary cohomology operation $\alpha : H ^ {n} ( \cdot ; \mathbf Z ) \rightarrow H ^ {n+} 3 ( \cdot ; \mathbf Z _ {2} )$ corresponding to the relation $S q ^ {2} \circ ( S q ^ {2} \circ \mathop{\rm mod} 2 ) = 0$. It enables one to classify mappings of $( n + 2 )$- dimensional polyhedra into the $n$- dimensional sphere, $n \geq 2$. The solution of the corresponding extension problem is as follows. Let $Y$ be an $( n + 3 )$- dimensional polyhedron and let a mapping $f : Y ^ {n+} 1 \rightarrow S ^ {n}$ be given, so that one has an element $y \in H ^ {n} ( Y ^ {n+} 1 )$, $y \in f ^ {*} ( s)$. Then a necessary condition for an extension of $f$ onto $Y ^ {n+} 2$ is that $S q ^ {2} ( z \mathop{\rm mod} 2 ) = 0$; if $f$ can be extended to $Y ^ {n+} 2$, then $\alpha$ is defined at $z$. It turns out that $f$ can be extended to $Y$ if and only if $0 \in \alpha ( z)$. Furthermore, $\alpha$ detects the mapping $S ^ {n+} 2 \rightarrow S ^ {n+} 1 \rightarrow S ^ {n}$ that is the composite of the suspension of the Hopf mapping $S ^ {3} \rightarrow S ^ {2}$ and defines a generator of the group $\pi _ {n+} 2 ( S ^ {n} ) = \mathbf Z _ {2}$.

The first solution of the "Hopf invariant problem55Q25Hopf invariant problem" (on the existence of elements of Hopf invariant one) was also given by means of secondary cohomology operations [7]. For a mapping $f : S ^ {2n-} 1 \rightarrow S ^ {n}$ the Hopf invariant $H ( f ) \in \mathbf Z$ is defined by the formula $u ^ {2} = H ( f ) v$, where $v \in H ^ {2n} ( S ^ {n} \cup f e ^ {2n} ) = \mathbf Z$, $u \in H ^ {n} ( S ^ {n} \cup fe ^ {2n} ) = \mathbf Z$ are generators. The oddness of $H ( f )$ is equivalent to the condition $S q ^ {n} u _ {n} \neq 0$, where $u _ {n} \in H ^ {n} ( S ^ {n} \cup f e ^ {2n} ; \mathbf Z _ {2} ) = \mathbf Z _ {2}$. When $n \neq 2 ^ {s}$, the operation $S q ^ {n}$ is decomposable in the class of primary operations, that is, $S q ^ {n} = \sum _ {t < n } a _ {t} S q ^ {t}$, so that $H ( f )$ can be odd only when $n = 2 ^ {s}$. But in the class of secondary operations, $S q ^ {2 ^ {s} }$ is decomposable when $s \neq 1 , 2 , 3$, and therefore $H ( f )$ can be odd only when $n = 2 , 4 , 8$.

In addition to the secondary cohomology operations there are tertiary ones, and more generally, higher cohomology operations of any order. Corresponding to a primary cohomology operation $\theta$ and an element $\phi \in H ^ {q} ( E ( \theta ) ; H )$ defining a secondary cohomology operation $\Phi$ there is a mapping $E ( \theta ) \rightarrow K ( H ; q )$ inducing from the Serre fibration over $K ( H ; q )$ the fibration $K ( H , q - 1 ) \rightarrow E ( \Phi ) \rightarrow E ( \theta )$; $E ( \Phi )$ is called the space of the cohomology operation $\Phi$. If one has a cohomology class $\gamma \in H ^ {r} ( E ( \Phi ) ; A )$, one can construct a tertiary cohomology operation $\Gamma : H ^ {n} ( X ; \pi ) \rightarrow H ^ {r} ( X ; A )$ defined on $\mathop{\rm Ker} \Phi$, the indeterminacy of which is $\mathop{\rm Im} ( i ^ {*} \gamma )$( under suitable restrictions on the dimension). Corresponding to this cohomology operation is the relation $\psi \circ \Phi = 0$, where $\psi$ is a primary cohomology operation, $1 _ \psi = i ^ {*} \gamma$. Inductive continuation of this process leads to the definition of an $n$- th order cohomology operation. In other words, given an $n$- th order cohomology operation $\xi$, the space of which is $E ( \xi )$, and an element $\lambda \in H ^ {s} ( E ( \xi ) , C )$, one constructs the $( n + 1 )$- th order cohomology operation $\Lambda$ defined on $\mathop{\rm Ker} \xi$. Moreover, the space $E ( \Lambda )$ is the space of the fibration induced from the Serre fibration over $K ( C , s )$ by the mapping $\lambda : E ( \xi ) \rightarrow K ( C , s )$. An axiom system for higher cohomology operations is constructed in [12].

The simplest examples of higher cohomology operations are the higher Bockstein homomorphisms. Let there be given a short exact sequence of groups

$$0 \rightarrow \mathbf Z \rightarrow ^ { p } \ \mathbf Z \rightarrow \mathbf Z _ {p} \rightarrow 0$$

so that

$$\dots \rightarrow H ^ {n} ( X) \rightarrow ^ { p } H ^ {n} ( X) \mathop \rightarrow \limits ^ { { \mathop{\rm mod}} p } \ H ^ {n} ( X ; \mathbf Z _ {p} ) \mathop \rightarrow \limits ^ \delta H ^ {n+} 1 ( X) \rightarrow \dots$$

is the corresponding exact sequence. Then the homomorphism $\beta = \mathop{\rm mod} p \circ \delta : H ^ {n} ( X ; \mathbf Z _ {p} ) \rightarrow H ^ {n+} 1 ( X ; \mathbf Z _ {p} )$ is also a Bockstein homomorphism; $\beta \in O _ {S} ( 1 ; \mathbf Z _ {p} , \mathbf Z _ {p} )$. The formula $\beta \circ \beta = 0$ holds; corresponding to this relation is the secondary cohomology operation $\beta _ {2}$. Furthermore, $\beta \circ \beta _ {2} = 0$, so that there is a tertiary cohomology operation $\beta _ {3}$. More generally, $\beta _ {r}$ is the $r$- th order cohomology operation constructed from the relation $\beta \circ \beta _ {r-} 1 = 0$. Here $\beta _ {r}$ is defined on $\mathop{\rm Ker} \beta _ {r-} 1$. An explicit description of $\beta _ {r}$ emerges in the following way: Let $x \in H ^ {n} ( X ; \mathbf Z _ {p} )$ and let $c$ be a cocycle with coefficients in $\mathbf Z _ {p}$ representing it. Then the equation $\beta _ {r-} 1 x = 0$ implies that there is an integral representative $z$ of the cocycle $c$, the coboundary $\delta z$ of which is divisible by $p ^ {r}$. Then $\beta _ {r} x$ is the cohomology class $\mathop{\rm mod} p$ of the cocycle $\delta z / p ^ {r}$. Thus, information about the action of the higher Bockstein cohomology operations in the groups $H ^ {*} ( X ; \mathbf Z _ {p} )$ enables one to calculate the free part and the $p$- component of the group $H ^ {*} ( X ; \mathbf Z )$.

To each partial cohomology operation $\Phi$ corresponds a homotopically-simple space $E ( \Phi )$ with a finite number of (non-trivial) homotopy groups. Conversely, one can associate with each space $E$ of this type a cohomology operation $\Phi$ for which $E$ and $E ( \Phi )$ are weakly homotopy equivalent, $E \sim ^ {w} E ( \Phi )$. For example, if $E$ is a space with two non-trivial homotopy groups $\pi _ {n} ( E) = \pi$, $\pi _ {m} ( E) = G$, $m > n$, then there is a mapping $E \rightarrow K ( \pi , n )$ inducing an isomorphism $\pi _ {n} ( E) \rightarrow \pi _ {n} ( K ( \pi , n ) )$. This mapping can be converted into a fibration with fibre $K ( G , m )$; this fibration is induced from the Serre fibration over $K ( G , m + 1 )$ by some mapping $K ( \pi , n ) \rightarrow K ( G , m + 1 )$; the latter defines a cohomology operation $\theta \in O ( n , m + 1 ; \pi , G )$.

These considerations enable one to describe the weak homotopy type of any space by associating with it the collection of higher cohomology operations $\{ k _ {n} \} _ {n=} 1 ^ \infty$, called its $n$- th Postnikov factors (see Postnikov system). For example, for the sphere $S ^ {n}$, $n > 3$, the first Postnikov factor is $S q ^ {2}$, and the second is $\alpha$.

Another important type of cohomology operations are the functional cohomology operations [3]. To define them, a mapping ( "function" ) $f : Y \rightarrow X$ and a cohomology operation $\theta \in O ( n , m ; \pi , G )$ are given. For $m \leq 2 - 2$, $\theta$ lies in the image of the cohomology suspension and is a group homomorphism. If $f$ is a closed imbedding (a cofibration) and $j : X \rightarrow ( X , Y )$ is the inclusion mapping, then the functional cohomology operation $\theta _ {f} : H ^ {n} ( X ; \pi ) \rightarrow H ^ {m-} 1 ( Y , G )$ is defined as the partial many-valued mapping

$$\theta _ {f} : H ^ {n} ( X ; \pi ) \leftarrow ^ { {j ^ {*}} } \ H ^ {n} ( X , Y ; \pi ) \mathop \rightarrow \limits ^ { {\theta ( X , Y ) }} \ H ^ {m} ( X , Y ; G ) \leftarrow ^ \delta$$

$$\leftarrow ^ \delta H ^ {m-} 1 ( Y ; G ) ;$$

$\theta _ {f} = \delta ^ {-} 1 \circ \theta \circ ( j ^ {*} ) ^ {-} 1$. This cohomology operation is defined on the subgroup $\mathop{\rm Ker} f ^ {*} \cap \mathop{\rm Ker} \theta _ {X}$ of $H ^ {n} ( X , \pi )$, and its indeterminacy is the subgroup $\mathop{\rm Im} ( {} ^ {1} \theta _ {y} ) + \mathop{\rm Im} ( f ^ {*} )$ of $H ^ {n-} 1 ( Y ; G )$. The construction of the functional cohomology operation is natural in $f$. Example. If for a mapping $f : Y \rightarrow X$ there exists a (primary) cohomology operation $\theta$ and a cohomology class $u$ of the space $X$ such that $\theta _ {f} ( u)$ is defined and $0 \notin \theta _ {f} ( u)$, then $f$ is essential. Functional and secondary cohomology operations are related to each other by the Peterson–Stein formulas (see [3]), enabling one in a number of cases to reduce the computation of secondary cohomology operations to that of primary and functional cohomology operations. There also exist higher functional cohomology operations [6]. The Massey product is a construction that is analogous to the higher cohomology operations in its structure and applications.

The concept of a cohomology operation has been carried over to generalized cohomology theories. A transformation $h ^ {n} ( X) \rightarrow h ^ {m} ( X)$( natural with respect to $X$) in a generalized cohomology theory $h ^ {*}$ is called a cohomology operation of type $( n , m )$. These cohomology operations from a group isomorphic to the group $h ^ {m} ( M _ {n} )$, where $\{ M _ {k} , s _ {k} \}$ is the $\Omega$- spectrum representing the theory $h ^ {*}$. The group of all stable cohomology operations is a ring $A ^ {h}$( with respect to composition), so that $h ^ {*}$ is an $A ^ {h}$- module natural with respect to $X$. The notions of a partial and a functional cohomology operation also have analogues in generalized cohomology theories.

By means of partial cohomology operations in ordinary cohomology theory one can solve, in principle, any homotopy problem; however, the practical application of a cohomology operation of order $n > 3$ is extremely laborious. At the same time, it often happens that a problem requiring for its solution ordinary cohomology operations of higher order can be easily solved by the application of primary cohomology operations in a suitably chosen generalized cohomology theory. For example, the "Hopf invariant problem" is easily solved by means of the Adams primary cohomology operations $\psi ^ {k}$ in $K$- theory [10]. These cohomology operations, introduced in [8] for the solution of vector fields on spheres, were the first examples of cohomology operations in a generalized cohomology theory.

The algebra $A ^ {h}$ was calculated [4] for $h = U ^ {*}$, the unitary cobordism theory, and was used in the construction of a spectral sequence of Adams type, the first term which is the cohomology space of the algebra $A ^ {U}$. Information on the action of the ring $A ^ {h}$ in the groups $h ^ {*} ( X)$ proves to be useful in the calculation of the Atiyah–Hirzebruch spectral sequence in the theory $h ^ {*}$.

#### References

 [1] L.S. Pontryagin, "A classification of mappings of a three-dimensional complex into the two-dimensional sphere" Mat. Sb. , 9 (1941) pp. 331–363 (In Russian) [2] N. Steenrod, "Products of cocycles and extensions of mappings" Ann. of Math. , 48 (1947) pp. 290–320 [3] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) [4] S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theory" Math. USSR.-Izv. , 31 (1967) pp. 827–913 Izv. Akad. Nauk SSSR Ser. Mat. (1967) pp. 855–951 [5] N.E. Steenrod, "Cohomology operations and obstructions to extending continuous functions" , Colloquium Lectures , Princeton Univ. Press (1957) [6] F.P. Peterson, "Functional cohomology operations" Trans. Amer. Math. Soc. , 86 (1957) pp. 197–211 [7] J.F. Adams, "On the non-existence of elements of Hopf invariant one" Ann. of Math. (2) , 72 (1960) pp. 20–104 [8] J.F. Adams, "Vector fields on spheres" Ann. of Math. , 75 (1962) pp. 603–632 [9] H. Cartan, "Algèbres d'Eilenberg–MacLane et homotopie" , Sem. H. Cartan , 7 (1954–1955) [10] M.F. Atiyah, "-theory: lectures" , Benjamin (1967) [11a] V.M. [V.M. Bukhshtaber] Buhštaber, "Modules of differentials of the Atiyah–Hirzebruch spectral sequence" Math. USSR-Sb. , 7 : 2 (1969) pp. 299–313 Mat. Sb. , 78 (1969) pp. 307–320 [11b] V.M. [V.M. Bukhshtaber] Buchštaber, "Modules of differentials of the Atiyah–Hirzebruch spectral sequence II" Math. USSR-Sb. , 12 (1970) pp. 59–75 Mat. Sb. , 83 (1970) pp. 61–76 [12] C.R.F. Maunder, "Cohomology operations of the -th kind" Proc. London Math. Soc. , 13 (1963) pp. 125–154

If, as is often done, the generalized cohomology theory defined by a spectrum $E$ is denoted by $E ^ {*}$, then the ring of stable cohomology operations of $E ^ {*}$ is $E ^ {*} ( E)$. The action of $E ^ {*} ( E)$ on $E ^ {*} ( X)$ is defined by assigning $gf : X \rightarrow E$ to $g : E \rightarrow E$ and $f : X \rightarrow E$. The ring $E ^ {*} ( E)$ in fact has a Hopf algebra structure. The dual Hopf algebra $E _ {*} ( E)$ is also a most useful object of equivalent power, and in fact is sometimes technically easier to work with [a4], [a5]. Cf. Generalized cohomology theories for more details and for the definition of the Hopf algebra structures on $E ^ {*} ( E)$ and $E _ {*} ( E)$. For complex cobordism ${M U } ^ {*}$( or $U ^ {*}$) and Brown–Peterson cohomology ${B P } ^ {*}$, the Hopf algebras ${M U } ^ {*} ( M U)$ and ${B P } ^ {*} ( B P)$ have interpretations in terms of the formal group laws defined by $M U$ and $B P$, cf. [a1], [a6].