Cohomology operation

A natural transformation of certain cohomology functors into others (most often — into themselves). By a cohomology operation of type , and being integers and Abelian groups, one means a family of mappings (not necessarily homomorphisms) between cohomology groups , defined for any space , such that for any continuous mapping (naturality). The set of all cohomology operations of type forms an Abelian group with respect to the composition: , and is denoted by .

Examples of cohomology operations. The Steenrod reduced powers and (cf. Steenrod reduced power); the Pontryagin square ; the Postnikov square; raising to the -th power, : for , where is a ring, , ; the Bockstein homomorphism ; cohomology operations induced by homomorphisms of the coefficient groups, for example, .

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 and be two spaces and let , be two elements. Does there exist a mapping such that ? A first sufficient condition for the absence of such an is the absence of a homomorphism with . (By this method, one can prove, for example, the Brouwer theorem on fixed points.) If exists, then the non-existence of can be established as follows: Let be a cohomology operation, , with , . Then , which is impossible. 2) Is the mapping: essential? Let . Then (for ) , . If there is a cohomology operation with , then is essential. In this case, the operation detects the mapping or the element .

There is an isomorphism of groups , where is an Eilenberg–MacLane space, and therefore (see Representable functor). The groups have been computed for all and and any finitely generated and [9].

The cohomology suspension of a cohomology operation is the mapping given by the composition

where is the suspension of . For example, , , . When , is an isomorphism. For any , is a group homomorphism.

By a stable cohomology operation of type and of degree one means a set with and . Such cohomology operations form an Abelian group , isomorphic to the group , the latter being the inverse limit of the sequence

The group is denoted by .

Examples of stable cohomology operations. The Steenrod powers and (where is a prime number), and the Bockstein homomorphism .

If and , then the cohomology operation is defined. In particular, one can define the composite of any two stable cohomology operations and , so that the group is a ring; is called the Steenrod algebra .

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

The extension theorem [2]: Let be an -dimensional polyhedron and let be its -dimensional skeleton. A mapping defines an element , where is a generator. This mapping can be extended to if and only if , where for the inclusion .

Corresponding to the cohomology operation , the mapping induces from the standard Serre fibration

the fibration

Secondary cohomology operations are cohomology classes of spaces . More precisely, let be given, where is an Abelian group. There exists for any with an element with , where is the mapping induced by ; the element depends on the choice of the element . The arbitrariness in the choice of is determined by the inverse image , that is, by the orbit of the action of the group on the set . When and , and are group homomorphisms, and, therefore, under a different choice of the element can change only by some element of the subgroup of the group . One defines the secondary cohomology operation by setting (the coset is uniquely determined by the element ). Thus, the mapping is defined on the subgroup and takes values in the quotient space , where is called the indeterminacy of the cohomology operation . An alternative terminology is that is a partial multi-valued cohomology operation from into .

Secondary cohomology operations are natural in the following sense: For any and any , one has [3]. If , then for some , so that , and therefore with zero indeterminacy. Here, , where is any cohomology operation in such that , and , so that the cohomology operation is a uni-valued cohomology operation restricted to .

To each secondary cohomology operation there corresponds a relation among the ordinary (primary) cohomology operations. If , then the cohomology operation is uniquely representable in the form with and . If is such that , then corresponding to the cohomology operation is the same relation . Conversely, there corresponds to any relation of the form a set of secondary cohomology operations , any two of which differ from each other by a primary cohomology operation defined on the kernel of .

A more general notion of a secondary cohomology operation is obtained by starting from a set with and the relation (see [3]).

Example of a secondary cohomology operation. Let

and let be a -generator. This gives rise to a secondary cohomology operation corresponding to the relation . It enables one to classify mappings of -dimensional polyhedra into the -dimensional sphere, . The solution of the corresponding extension problem is as follows. Let be an -dimensional polyhedron and let a mapping be given, so that one has an element , . Then a necessary condition for an extension of onto is that ; if can be extended to , then is defined at . It turns out that can be extended to if and only if . Furthermore, detects the mapping that is the composite of the suspension of the Hopf mapping and defines a generator of the group .

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 the Hopf invariant is defined by the formula , where , are generators. The oddness of is equivalent to the condition , where . When , the operation is decomposable in the class of primary operations, that is, , so that can be odd only when . But in the class of secondary operations, is decomposable when , and therefore can be odd only when .

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 and an element defining a secondary cohomology operation there is a mapping inducing from the Serre fibration over the fibration ; is called the space of the cohomology operation . If one has a cohomology class , one can construct a tertiary cohomology operation defined on , the indeterminacy of which is (under suitable restrictions on the dimension). Corresponding to this cohomology operation is the relation , where is a primary cohomology operation, . Inductive continuation of this process leads to the definition of an -th order cohomology operation. In other words, given an -th order cohomology operation , the space of which is , and an element , one constructs the -th order cohomology operation defined on . Moreover, the space is the space of the fibration induced from the Serre fibration over by the mapping . 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

so that

is the corresponding exact sequence. Then the homomorphism is also a Bockstein homomorphism; . The formula holds; corresponding to this relation is the secondary cohomology operation . Furthermore, , so that there is a tertiary cohomology operation . More generally, is the -th order cohomology operation constructed from the relation . Here is defined on . An explicit description of emerges in the following way: Let and let be a cocycle with coefficients in representing it. Then the equation implies that there is an integral representative of the cocycle , the coboundary of which is divisible by . Then is the cohomology class of the cocycle . Thus, information about the action of the higher Bockstein cohomology operations in the groups enables one to calculate the free part and the -component of the group .

To each partial cohomology operation corresponds a homotopically-simple space with a finite number of (non-trivial) homotopy groups. Conversely, one can associate with each space of this type a cohomology operation for which and are weakly homotopy equivalent, . For example, if is a space with two non-trivial homotopy groups , , , then there is a mapping inducing an isomorphism . This mapping can be converted into a fibration with fibre ; this fibration is induced from the Serre fibration over by some mapping ; the latter defines a cohomology operation .

These considerations enable one to describe the weak homotopy type of any space by associating with it the collection of higher cohomology operations , called its -th Postnikov factors (see Postnikov system). For example, for the sphere , , the first Postnikov factor is , and the second is .

Another important type of cohomology operations are the functional cohomology operations [3]. To define them, a mapping ( "function" ) and a cohomology operation are given. For , lies in the image of the cohomology suspension and is a group homomorphism. If is a closed imbedding (a cofibration) and is the inclusion mapping, then the functional cohomology operation is defined as the partial many-valued mapping

. This cohomology operation is defined on the subgroup of , and its indeterminacy is the subgroup of . The construction of the functional cohomology operation is natural in . Example. If for a mapping there exists a (primary) cohomology operation and a cohomology class of the space such that is defined and , then 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 (natural with respect to ) in a generalized cohomology theory is called a cohomology operation of type . These cohomology operations from a group isomorphic to the group , where is the -spectrum representing the theory . The group of all stable cohomology operations is a ring (with respect to composition), so that is an -module natural with respect to . 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 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 in -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 was calculated [4] for , 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 . Information on the action of the ring in the groups proves to be useful in the calculation of the Atiyah–Hirzebruch spectral sequence in the theory .

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

Comments

The spectral sequence introduced in [4], often called the Adams–Novikov spectral sequence, has been the basis for much subsequent work; see [a1].

If, as is often done, the generalized cohomology theory defined by a spectrum is denoted by , then the ring of stable cohomology operations of is . The action of on is defined by assigning to and . The ring in fact has a Hopf algebra structure. The dual Hopf algebra 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 and . For complex cobordism (or ) and Brown–Peterson cohomology , the Hopf algebras and have interpretations in terms of the formal group laws defined by and , cf. [a1], [a6].

References

[a1]	D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 269–276; 429–432
[a3]	N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[a4]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapts. 17; 18
[a5]	J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974)
[a6]	P.S. Landweber, " and typical formal groups" Osaka J. Math. , 12 (1975) pp. 499–506