Homotopy type

A class of homotopy-equivalent topological spaces. Two mappings $ f: \ X \rightarrow Y $ and $ g: \ Y \rightarrow X $ are said to be mutually-inverse homotopy equivalences if $ f \circ g \sim 1 _{Y} $ and $ g \circ f \sim 1 _{X} $ . If only the first condition is met, $ g $ is said to be a homotopy monomorphism and $ f $ is said to be a homotopy epimorphism. A mapping is a homotopy equivalence if and only if it is both a homotopy monomorphism and a homotopy epimorphism. If there exists a homotopy epimorphism $ f: \ X \rightarrow Y $ , then one says that $ Y $ dominates $ X $ . If there exists a homotopy equivalence $ f: \ X \rightarrow Y $ , then $ X $ and $ Y $ are said to be homotopy equivalent, or spaces of the same homotopy type.

The problem of homotopy type consists in finding necessary and sufficient conditions for homotopy equivalence of arbitrary spaces. It may be convenient to weaken this statement somewhat. A mapping $ f: \ X \rightarrow Y $ is said to be a weak homotopy equivalence if it induces an isomorphism of the homotopy groups in all dimensions (cf. Homotopy group). Correspondingly, two spaces $ X $ and $ Y $ are said to be weakly homotopy equivalent if there exists either a weak homotopy equivalence $ X \rightarrow Y $ or a weak homotopy equivalence $ Y \rightarrow X $ . Since any homotopy equivalence is a weak homotopy equivalence, it follows that homotopy-equivalent spaces are weakly homotopy equivalent. The converse is true if the spaces are CW-complexes (Whitehead's theorem, cf. CW-complex). This theorem is based on the facts that: 1) a mapping $ f: \ X \rightarrow Y $ is a homotopy equivalence if and only if $ X $ is a deformation retract of the cylinder (cf. Mapping cylinder) $ M _{f} $ of $ f $ ; 2) a mapping $ f: \ X \rightarrow Y $ is a weak homotopy equivalence if and only if the subspace $ X $ of the cylinder $ M _{f} $ is homotopy representative (cf. Representative subspace); and 3) a subdivision of a $ \mathop{\rm CW}\nolimits $ - complex is representative if and only if it is a deformation retract.

Thus, the problem of homotopy types on the category of $ \mathop{\rm CW}\nolimits $ - complexes is equivalent with the problem of weak homotopy types. On the other hand, any space $ X $ is weakly homotopy equivalent to the geometric realization of its singular simplicial set $ S(X) $ . For this reason, the problem of weak homotopy types need be considered, without restricting generality, only for $ \mathop{\rm CW}\nolimits $ - complexes.

Two mappings $ f,\ g : \ X \rightarrow Y $ are said to be $ n $ - homotopic if for any $ n $ - complex $ \mathop{\rm CW}\nolimits $ of dimension $ K $ and any mapping $ \leq n $ the mappings $ \phi : \ K \rightarrow X $ and $ f \circ \phi $ are homotopic. If $ g \circ \phi $ is a $ X $ - complex, this holds if and only if $ \mathop{\rm CW}\nolimits $ . Spaces that are equivalent with respect to $ f | _ {X ^{n}} \sim g | _ {X ^{n}} $ - homotopy are called spaces of the same $ n $ - homotopy type. Two $ n $ - complexes $ \mathop{\rm CW}\nolimits $ and $ K $ are said to be complexes of the same $ L $ - type (denoted by $ n $ ) if their $ K \sim _{n} L $ - th skeletons $ n $ and $ K ^{n} $ have the same $ L ^{n} $ - homotopy type. If $ (n - 1 ) $ , then $ K \sim _{n} L $ for any $ K \sim _{m} L $ . This remains true even for $ m \leq n $ , if the $ n = \infty $ - type is understood to be the homotopy type. In other words, the concept of being of $ \infty $ - type is homotopy invariant. The importance of the concept of being of $ n $ - type for the problem of homotopy types is due to the fact that two $ n $ - dimensional $ \mathop{\rm CW}\nolimits $ - complexes are homotopy equivalent if and only if they have the same $ (n + 1 ) $ - type.

Let $ X $ be an arbitrary space (for the sake of simplicity — a totally-connected space). A simplicial subset $ M (X) $ of the simplicial set $ S(X) $ is said to be minimal if it contains all the singular simplices that are mappings into some fixed point $ x _{0} \in X $ , and if for any simplex $ \sigma \in S (X) $ with all faces in $ M(X) $ there exists a unique simplex in $ M(X) $ that is homotopic to $ \sigma $ ( with respect to the boundary of the standard simplex). Minimal subsets exist and, up to an isomorphism, are uniquely determined by $ X $ . Moreover, two spaces are weakly homotopy equivalent if and only if their minimal simplicial sets are isomorphic. Thus, in order to solve the problem of weak homotopy type, all that remains to be done is to find a satisfactory description of the simplicial sets $ M(X) $ .

Let $ \Delta _{q} $ be a $ q $ - dimensional standard simplex, considered as a simplicial decomposition (with respect to its standard triangulation), and let $ C ^{n} ( \Delta _{q} ,\ \pi ) $ be its $ n $ - dimensional cochain group over an Abelian group $ \pi $ ( more exactly, the group of normalized $ n $ - dimensional cochains of the simplicial set $ O( \Delta _{q} ) $ , cf. Cochain). Let $ E ( \pi ,\ n) $ be a simplicial set in which the simplices of dimension $ q $ are cochains in $ C ^{n} ( \Delta _{q} ,\ \pi ) $ , and the boundary operators $ \partial _{i} $ and degeneracies $ s _{i} $ are cochain mappings induced by the standard simplicial mappings $ e _{i} : \ O ( \Delta _{q-1} ) \rightarrow O ( \Delta _{q} ) $ and $ f: \ O ( \Delta _{q+1} ) \rightarrow O ( \Delta _{q} ) $ ( the mapping $ e _{i} $ " makes free" the $ i $ - th vertex, while the mapping $ f _{i} $ " glues" the $ i $ - th and the $ (i + 1) $ - th vertices). The simplices that are cocycles form a certain simplicial subset $ K ( \pi ,\ n) $ in $ E ( \pi ,\ n) $ . The coboundary operator $ \delta : \ C ^{n} ( \Delta _{q} ,\ \pi ) \rightarrow C ^{n+1} ( \Delta _{q} ,\ \pi ) $ defines a simplicial mapping $ \delta : \ E ( \pi ,\ n) \rightarrow K ( \pi ,\ n + 1) $ , whose kernel is $ K ( \pi ,\ n) $ . The mapping $ \delta $ is a fibration (in the sense of Kan) with fibre $ K ( \pi , n) $ . In addition, the simplicial set $ K ( \pi ,\ n) $ is an object of type $ K ( \pi ,\ n) $ in the category of simplicial sets (with respect to homotopy groups in the sense of Kan, see Eilenberg–MacLane space), while the simplicial set $ E ( \pi ,\ n ) $ is homotopy trivial (homotopy equivalent to a "point" ). Thus, the fibration $ \delta $ is a simplicial analogue of the Serre fibration of paths over a space of type $ K ( \pi ,\ n + 1) $ .

The simplicial set $ K ( \pi ,\ n) $ for $ n = 1 $ is also meaningful for any (not necessarily Abelian) group $ \pi $ . The simplicial set $ K ( \pi ,\ 1 ) $ thus obtained is the standard simplicial resolution of $ \pi $ .

Let $ \pi _{1} $ be a multiplicative group of operators on the additive group $ \pi $ . Let $ N $ be an arbitrary simplicial set over $ \pi _{1} $ . Let $ a $ be a cocycle in $ N $ . In the group of cocycles of this set over $ \pi $ there is defined a coboundary operator $ \delta _{a} $ with respect to $ a $ . Let $ \sigma $ be an arbitrary $ q $ - dimensional simplex in $ N $ and let $ t _ \sigma : \ O ( \Delta _{q} ) \rightarrow N $ be its characteristic mapping (cf. Simplicial set). This defines a cocycle $ t _ \sigma ^{*} (a) $ in $ O ( \Delta _{q} ) $ . Let the coboundary operator with respect to this cocycle be denoted by the symbol $ \delta _ {a, \sigma} $ . Let $ k ^{n+1} $ be an arbitrary $ (n + 1 ) $ - dimensional cocycle of the simplicial set $ N $ over $ \pi $ with respect to $ a $ . If in the direct product of the simplicial sets $ N $ and $ E ( \ \pi , n) $ one considers the subset $ P = P (N,\ k ^{n+1} ) $ consisting of all possible pairs $ ( \sigma ,\ u) $ , $ \sigma \in N $ , $ u \in E ( \pi ,\ n) $ , for which $ t _ \sigma ^{*} ( k ^{n+1} ) = \delta _ {a, \sigma} u $ , then $ P $ is a simplicial subset. The formula $ p ( \sigma ,\ u) = \sigma $ defines a surjective mapping $ p: \ P \rightarrow N $ , which is a fibration (in the sense of Kan). This fibration will be denoted by $ p (N,\ k ^{n+1} ) $ . If the cocycle $ a $ is trivial, then this fibration is identical with the fibration induced by the simplicial mapping $ N \rightarrow K ( \pi ,\ n + 1) $ which corresponds to the cocycle $ k ^{n+1} $ in the fibration $ E ( \pi ,\ n) \mapsto K ( \pi ,\ n + 1) $ . The fibration $ p $ over an $ n $ - dimensional skeleton $ N ^{n} $ of the simplicial set $ N $ has a section $ \sigma \rightarrow ( \sigma ,\ 0) $ and $ k ^{n+1} $ is an obstruction to an extension of this section to $ N ^{n+1} $ . After identification of the simplices $ \sigma $ and $ ( \sigma ,\ 0) $ it follows that $ N ^{n} \subset P ^{n} $ . Moreover, $ N ^{n-1} = P ^{n-1} $ .

Consider the following sequence of fibrations of simplicial sets:$$ \tag{1} \dots \rightarrow P _{n+1} \stackrel{ {p _{n}}} \rightarrow P _{n} \rightarrow \dots \rightarrow P _{2} \stackrel{ {p _{1}}} \rightarrow P _{1} . $$ Its initial term $ P _{1} $ is the simplicial set $ K ( \pi _{1} ,\ 1) $ constructed from the multiplicative group $ \pi _{1} $ . By definition, one-dimensional simplices of $ P _{1} $ are in natural bijective correspondence with the elements of $ \pi _{1} $ . Putting such a simplex into correspondence with the corresponding element of $ \pi _{1} $ yields some one-dimensional cocycle $ a _{1} $ over $ \pi _{1} $ in $ P _{1} $ . Let the cocycle $ a _{n} $ be defined in $ P _{n} $ by the inductive formula $ a _{n} = p _{n-1} ^{*} ( a _{n-1} ) $ . The sequence (1) is called a homotopy resolvent, or a Postnikov system (the original name was: natural system), if for any $ n \geq 1 $ the simplicial set $ P _{n+1} $ is a set of type $ P ( P _{n} ,\ k _{n} ^{n+2} ) $ , where $ k _{n} ^{n+2} $ is a cocycle of dimension $ n + 2 $ in $ P _{n} $ over some $ \pi _{1} $ - group $ \pi _{n+1} $ with respect to the cocycle $ a _{n} $ ( and the fibration $ p _{n} $ is the fibration $ p ( P _{n} ,\ k _{n} ^{n+2} ) $ ). This sequence is called the resolution of the simplicial set $ M $ if for any $ n \geq 1 $ a simplicial mapping $ q _{n} : \ M \rightarrow P _{n} $ is given which is an isomorphism on $ M ^{n} $ and which is such that $ p _{n} \circ q _{n+1} = q _{n} $ . The resolution uniquely determines the simplicial set $ M $ up to an isomorphism. On the other hand, the resolution itself is uniquely determined by the groups $ \pi _{1} ,\ \pi _{2} \dots $ and the cocycles $ k _{1} ^{3} \dots k _{n} ^{n+2} , . . . $ . For this reason, an object $ \{ \pi _{n} ,\ k _{n} ^{n+2} \} $ , consisting of groups $ \pi _{n} $ and cocycles $ k _{n} ^{n+2} $ , may also be called a resolution.

Not all simplicial sets $ M $ have a resolution. According to the basic theorem in the theory of homotopy resolutions, a simplicial set $ M $ has a resolution if and only if it is isomorphic to the minimal simplicial set of a certain topological space $ X $ . In this case $ \pi _{n} = \pi _{n} (X) $ .

The resolution of the minimal set $ M = M(X) $ is constructed as follows. Let $ \sigma $ be an arbitrary $ q $ - dimensional simplex in $ M(X) $ . This simplex represents a mapping $ \sigma : \ \Delta _{q} \rightarrow X $ that maps all vertices of the simplex $ \Delta _{q} $ to the point $ x _{0} $ . For this reason it defines some element of $ \pi _{1} = \pi _{1} (X,\ x _{0} ) $ on any one-dimensional boundary of $ \Delta _{q} $ . Thus appears on $ \Delta _{q} $ a one-dimensional cocycle over $ \pi $ , i.e. a $ q $ - dimensional simplex in $ P _{1} = K ( \pi _{1} ,\ 1) $ ; it will be denoted by $ q _{1} ( \sigma ) $ . In this way one obtains a certain (simplicial) mapping $ q _{1} : \ M \rightarrow P _{1} $ . This mapping is an isomorphism on $ M ^{1} $ and an epimorphism on $ M ^{2} $ . The next step is induction: Let the simplicial set $ P _{n} $ and the simplicial mapping $ q _{n} : \ M \rightarrow P _{n} $ , which is an isomorphism on $ M ^{n} $ and an epimorphism on $ M ^{n+1} $ , have been already constructed for some $ n \geq 1 $ . The mapping $ q _{n} \mid _ {M ^{n+1}} $ has a right inverse $ q _{n} ^ \prime : \ P _{n} ^{n+1} \rightarrow M ^{n+1} $ . Let $ k _{n} ^{n+2} $ be an obstruction to extension of this mapping to $ M ^{n+2} $ . The obstruction $ k _{n} ^{n+2} $ is an $ (n + 2) $ - dimensional cocycle in $ P _{n} $ over the group $ \pi _{n+1} = \pi _{n+1} (X) $ with respect to $ a _{n} $ . For any $ (n + 1 ) $ - dimensional simplex $ \tau $ in $ M $ the simplex $ q _{n} ^ \prime q _{n} ( \tau ) = \tau ^ \prime $ is compatible with $ \tau $ , and for this reason the difference element $ d ( \tau ,\ \tau ^ \prime ) \in \pi _{n+1} $ is defined (cf. Difference element in $ K $ - theory). Let $ \sigma $ be an arbitrary $ q $ - dimensional simplex in $ M $ . On each $ (n + 1 ) $ - dimensional boundary of the simplex $ \Delta _{q} $ it defines a certain $ (n + 1) $ - dimensional simplex $ \tau $ . Assigning to this face the element $ d ( \tau ,\ \tau ^ \prime ) $ yields a certain $ (n + 1) $ - dimensional cochain in $ \Delta _{q} $ over $ \pi _{n+1} $ , i.e. some $ q $ - dimensional simplex $ r _{n} ( \sigma ) $ of the simplicial set $ E ( \pi _{n+1} ,\ n + 1 ) $ . The pair $ q _{n+1} ( \sigma ) = ( q _{n} ( \sigma ) ,\ r _{n} ( \sigma ) ) $ belongs to the simplicial set $ P _{n+1} = P ( P _{n} ,\ k _{n} ^{n+2} ) $ . To finish the induction one has to observe that the constructed mapping $ q _{n+1} : \ M \rightarrow P _{n+1} $ is a simplicial isomorphism on $ M ^{n+1} $ and an epimorphism on $ M ^{n+2} $ .

The resolution $ \{ \pi _{n} ,\ k _{n} ^{n+2} \} $ is not uniquely constructed from the set $ M(X) $ : there is freedom in the selection of the inverse mappings $ q _{n} ^ \prime $ . The simplest way of describing this non-uniqueness is to consider resolutions in the sense of (1). In fact, two such resolutions $ \{ \bar{P} _{n} ,\ \bar{p} _{n} \} $ and $ \{ P _{n} ,\ p _{n} \} $ result from the same minimal simplicial set $ M(X) $ if and only if they are isomorphic as sequences of mappings, i.e. if for any $ n \geq 1 $ there exists an isomorphism $ \theta _{n} : \ P _{n} \rightarrow \bar{P} _{n} $ such that $ \theta _{n} \circ p _{n} = \bar{p} _{n} \circ \theta _{n+1} $ . In order to describe such an isomorphism in terms of the resolutions $ \{ \pi _{n} ,\ k _{n} ^{n+2} \} $ and $ \{ \bar{p} _{n} ,\ \bar{k} {} _{n} ^{n+2} \} $ it should be noted that the existence of an isomorphism $ \theta _{1} : \ P _{1} \rightarrow \bar{P} _{1} $ is equivalent to the existence of an isomorphism of groups $ \Theta _{1} : \ \pi _{1} \rightarrow \bar \pi _{1} $ . Here $ \theta _{1} ^{*} ( \bar{a} _{1} ) = \Theta _{1} (a _{1} ) $ . Furthermore, for the isomorphism $ \theta _{n} : \ P _{n} \rightarrow \bar{P} _{n} $ there is a subsequent isomorphism $ \theta _{n+1} : \ P _{n+1} \rightarrow \bar{P} _{n+1} $ if and only if there exists a $ \Theta _{1} $ - isomorphism $ \Theta _{n+1} : \ \pi _{n+1} \rightarrow \bar \pi _{n+1} $ ( see Operator homomorphism) and a cochain $ l ^{n+1} \in C ^{n+1} (P _{n} ,\ \bar \pi _{n+1} ) $ such that$$ \tag{2} \theta _{n} ^{*} {( \bar{k} {} _{n} ^ {n + 2} )} - \Theta _ {n + 1} (k _{n} ^ {n + 2} ) = \delta _ {\theta _{n} ^{*} (a _{n} )} l ^ {n + 1} . $$ Here the isomorphism $ \Theta _{n+1} $ is defined by the formula$$ \tag{2'} \Theta _ {n + 1} ( \sigma ,\ u) = ( \sigma ,\ u - t _ \sigma ^{*} (l ^ {n + 1} )). $$ Resolutions $ \{ \pi _{n} ,\ k _{n} ^{n+2} \} $ and $ \{ \bar \pi _{n} ,\ {\bar{k} {} _{n} ^{n+2}} \} $ arise from the same simplicial set if and only if there exists an isomorphism $ \theta _{n} : \ \pi _{n} \rightarrow \bar \pi _{n} $ that is a $ \theta _{1} $ - isomorphism for $ n > 1 $ , such that for any $ n \geq 1 $ relation (2) holds, where $ \Theta _{n} $ is the isomorphism subsequently defined by (2') for $ n > 1 $ , while for $ n = 1 $ it is the isomorphism induced by $ \theta _{1} $ . In this case the resolutions $ \{ \pi _{n} ,\ k _{n} ^{n+2} \} $ and $ \{ \bar \pi _{n} ,\ {\bar{k} {} _{n} ^{n+2}} \} $ are called isomorphic. The resolution of the symplicial set $ M (X) $ is called the homotopy resolution of the space $ X $ . Summarizing, two spaces are weakly homotopy equivalent if and only if their homotopy resolutions are isomorphic; in particular, two $ \mathop{\rm CW}\nolimits $ - complexes are homotopy equivalent if and only if their homotopy resolutions are isomorphic.

If (2) is satisfied only for $ n < m $ , then the isomorphisms $ \theta _{n} $ only exist for $ n \leq m $ . In this situation one says that the given resolutions are $ m $ - isomorphic. Two $ \mathop{\rm CW}\nolimits $ - complexes are of the same $ n $ - type if and only if their homotopy resolutions are $ (n - 1) $ - isomorphic.

The given solution to the problem of homotopy types (or $ n $ - types) makes it possible to demonstrate a series of general theorems and to essentially clarify the principal sides of the subject (but an explicit computation of resolutions is only possible in a few cases). It follows that for any simply-connected space with finite homology groups the homotopy groups can be effectively computed. An analogous statement holds for spaces whose homology groups are only finitely generated [2]. The fact that the homotopy type is completely determined by the resolution shows that any problem in homotopy theory reduces to some statement on the resolutions of the corresponding spaces. This makes it possible to classify problems by the number of cycles $ k _{n} ^{n+2} $ which form part of their solutions. If the space being studied is $ (m - 1) $ - connected, then its resolution starts in fact with the term $ P _{n} = K ( \pi _{n} ,\ n) $ . If the solution of a given problem can be formulated in terms of the first non-trivial group $ \pi _{n} $ only, this problem is called a problem of order zero (e.g. the Hopf–Whitney problem on the classification of mappings of an $ n $ - dimensional polyhedron into an $ (n - 1) $ - connected space). If the groups $ \pi _{n} $ , $ \pi _{n+1} $ and the cocycle $ k _{n} ^{n+2} $ are used, the problem is called a problem of order one (e.g. the problem of classifying mappings from an $ (n + 1) $ - dimensional polyhedron into an $ (n - 1) $ - connected space). Similarly one defines problems of order two, three, etc. Effective solutions of problems of order zero or one are known. This is related to the fact that for any $ (n - 1) $ - connected space the homology class of the cocycle $ k _{n} ^{n+2} $ can be effectively computed; it has the form $ Sq _ \eta ^{2} \iota $ , where $ \iota $ is the fundamental class of the space $ K ( \pi _{n} ,\ n) $ and $ Sq _ \eta ^{2} $ , for $ n \geq 2 \iota $ is the Steenrod operation corresponding to the natural pairing $ \eta : \ \pi _{n} \otimes \pi _{n} \rightarrow \pi _{n+1} $ , and for $ n = 2 $ some invariant of it, $ \iota $ is known as the Pontryagin square. For problems of higher orders it is necessary to effectively compute the next cocycles $ k _{n+1} ^{n+3} ,\ k _{n+2} ^{n+4} , . . . $ . Each of these cocycles is obtained from the fundamental class by some cohomology operation of corresponding order. This, in particular, shows that the solution of any problem in homology theory can be formulated in terms of cohomology operations. However, because of the great complexity of higher-order operations, only solutions to special problems of a higher order have been obtained, using considerations of special character. Some general progress has been achieved under the assumption of stability: Under this assumption sufficiently far computation of the differentials of the Adams spectral sequence is equivalent to the computation of some stable operation of high order.

The theory of homotopy resolutions can be re-formulated in the following "geometric" form. An arbitrary sequence of fibrations in the sense of Serre,$$ \tag{3} X _ {n + 1} \stackrel{ {p _{n}}} \rightarrow X _{n} \rightarrow \dots \rightarrow X _{2} \stackrel{ {p _{1}}} \rightarrow X _{1} , $$ in which each space $ X _{n} $ has the property that $ \pi _{m} (X _{n} ) = 0 $ for $ m > n $ , is called a resolution. This sequence is called the resolution of a space $ X $ if for any $ n \geq 1 $ a mapping $ q _{n} : \ X \rightarrow X _{n} $ is defined that induces an isomorphism of the homotopy groups in dimension $ n $ , and that is such that $ p _{n} \circ q _{n+1} = q _{n} $ . This resolution is uniquely determined (up to an isomorphism, cf. Sequence category) by the groups $ \pi _{n} = \pi _{n} (X) $ and the characteristic classes $ k _{n} ^{n+2} $ of the fibration $ p _{n} $ . The resolution exists for any totally-connected space $ X $ ( such as the geometric realization of the "algebraic" resolution (1)) and determines this space up to weak homotopy equivalence. A fibre of the fibration $ p _{n} : \ X _{n+1} \rightarrow X _{n} $ is a space of type $ K ( \pi _{n+1} ,\ n + 1) $ and, if $ X $ is homotopy $ n $ - simple (cf. Homotopy group), e.g. simply-connected, this fibration is induced by the Serre fibration of paths over the space $ K ( \pi _{n+2} ,\ n + 2) $ using a mapping $ X _{n} \rightarrow K ( \pi _{n+1} ,\ n + 2) $ representing the cohomology class $ k _{n} ^{n+2} $ ( cf. Eilenberg–MacLane space; Representable functor).

If the space $ X $ is $ (n - 1) $ - connected, then its resolution actually starts with $ X _{n} = K ( \pi _{n} ,\ n) $ . For $ n > 1 $ it is convenient to regard, alongside with the "absolute" resolution (3), the resolution modulo a prime number $ p $ , the definition of which differs from the definition of (3) only in that the homotopy groups are replaced by their $ p $ - components. If the resolutions modulo any prime number $ p $ have been found for $ X $ , then the determination of its "absolute" resolution presents no difficulties. Therefore, in problems of computing resolutions (including the computation of the homotopy groups) one usually restricts to resolutions "modulo" , in the computation of which the powerful methods of the theory of spectral sequences and of homology operations can be used (cf. Spectral sequence). For some spaces the computation of the resolution has been advanced sufficiently far.

For example, for the sphere $ S ^{n} $ ( for large $ n $ so that the stability condition is fulfilled) sufficiently many terms of its resolution modulo 2 are known. It is sufficient to describe the groups $ \pi _{n+r} $ ( i.e. the $ 2 $ - components of the group $ \pi _{n+r} (S ^{n} ) $ ) and the cohomology classes $ k _{n+r} ^{n+r+2} $ . The first groups $ \pi _{n+r} $ have the following form:

<tbody> </tbody> $ r $ 0 1 2 3 4 5 6 7 $ \pi _{n+r} $ $ \mathbf Z $ $ \mathbf Z _{2} $ $ \mathbf Z _{2} $ $ \mathbf Z _{8} $ $ 0 $ $ 0 $ $ \mathbf Z _{2} $ $ \mathbf Z _{16} $

The class $ k _{n} ^{n+2} $ has the form $ Sq ^{2} \iota \in H ^{n+2} ( \mathbf Z ,\ n; \ \mathbf Z _{2} ) $ , where $ \iota \in H ^{n} ( \mathbf Z ,\ n; \ \mathbf Z _{2} ) $ is the fundamental class. The next class $ k _{n+1} ^{n+3} \in H ^{n+3} (K _{1} ; \ \mathbf Z _{2} ) $ has the property that on the fibre $ K ( \mathbf Z _{3} ,\ n + 1) $ of the fibration $ p _{n} $ it cuts out the class $ Sq ^{2} \iota _{n+1} \in H ^{n+3} ( \mathbf Z _{2} ,\ n + 1; \ \mathbf Z _{2} ) $ , and it is uniquely determined by it. Similarly, the class $ k _{n+2} ^{n+4} \in H ^{n+4} (X _{2} ; \ \mathbf Z _{8} ) $ is uniquely characterized by the fact that, by reducing modulo 2, it transforms to the class $ Sq ^{4} \iota \in H ^{n+4} (X _{2} ; \ \mathbf Z _{2} ) $ . The classes $ k _{n+3} ^{n+5} $ and $ k _{n+4} ^{n+6} $ vanish and the class $ k _{n+5} ^{n+7} \in H ^{n+7} (X _{n+5} ; \ \mathbf Z _{2} ) $ is uniquely characterized by the fact that on the fibre $ K ( \mathbf Z _{8} ,\ n + 3) $ of the fibration $ p _{n+2} $ it cuts out the class $ Sq ^{4} \iota _{n+3} \in H ^{n+7} ( \mathbf Z _{8} ,\ n + 3; \ \mathbf Z _{2} ) $ . Finally, $ k _{n+6} ^{n+8} \in H ^{n+8} (X _{n+6},\ \mathbf Z _{16} ) $ is characterized by the fact that, by reducing modulo 2, it transforms to the class $ Sq ^{8} \iota \in H ^{n+8} (X _{n+6} ; \ \mathbf Z _{2} ) $ .

References

[1]	M.M. Postnikov, "Studies on the homotopy theory of continuous mappings" Trudy Mat. Inst. Steklov. , 46 (1955) (In Russian) (1. Algebraic system theory; 2. The natural system and homotopy theory)
[2]	E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20
[3]	R.E. Mosher, M.C. Tangora, "Cohomology operations and their applications in homotopy theory" , Harper & Row (1968) pp. Chapt. 13

Comments

The article above concentrates on one aspect of homotopy types, viz. Postnikov towers (or Postnikov decompositions) and gives a great deal of detail concerning the state of affairs in the 1950's.

According to [a4] one should distinguish three directions in homotopy theory: suspension theory, which separates stable phenomena from those which are not (cf. Suspension, and also Cohomology operation and Homotopy group), explicit geometric constructions such as Whitehead products (cf. Whitehead product) and the method of killing homotopy groups. The Postnikov tower arises by killing successively homotopy groups above a given dimension. The "dual" construction involves killing at each stage all the homotopy groups below a given dimension. This yields the Whitehead tower$$ \begin{array}{rcl} {} &{} &\cdot \\ {} &{} &\cdot \\ {} &{} &\cdot \\ {} &{} &\downarrow \\ K ( \pi _{n} ,\ n - 1 ) &\rightarrow &X _{n} \\ {} &{} &\downarrow \\ {} &{} &X _{n-1} \\ {} &{} &\downarrow \\ {} &{} &\cdot \\ {} &{} &\cdot \\ {} &{} &\cdot \\ {} &{} &\downarrow \\ K ( \pi _{1} ,\ 0 ) &\rightarrow &X _{1} \\ {} &{} &\downarrow \\ {} &{} & X \\ \end{array} $$ where $ X _{n} $ is $ n $ - connected, i.e. $ \pi _{q} ( X _{n} ) = 0 $ for all $ q \leq n $ , the fibration $ X _{n} \rightarrow X _{n-1} $ has as fibre the Eilenberg–MacLane space $ K ( \pi _{n} ,\ n - 1 ) $ ( where $ \pi _{n} = \pi _{n} (X) $ ), and where above dimension $ n $ the homotopy groups of $ X _{n} $ and $ X $ agree.

In contrast, the Postnikov approximation (Postnikov tower) of a connected CW-complex is a sequence of fibrations$$ \begin{array}{cccc} {} & &Y _{n} & \leftarrow K ( \pi _{n} ,\ n) \\ {} & \nearrow ^ {q _{n}} &\downarrow _ {p _{n-1}} &{} \\ {} &{} &\cdot &{} \\ {} &{} &\cdot &{} \\ {} &{} &\cdot &{} \\ {} &{} &\downarrow _ {p _{2}} &{} \\ {} &{} &Y _{2} & \leftarrow K ( \pi _{2} ,\ 2) \\ {} & \nearrow ^ {q _{2}} &\downarrow _ {p _{1}} &{} \\ X & \rightarrow _ {q _{1}} &Y _{1} & = K( \pi _{1} ,\ 1 ) \\ \end{array} $$ with the fibres of $ Y _{n} \rightarrow Y _{n-1} $ equal to $ K ( \pi _{n} ,\ n ) $ , $ p _{n-1} \circ q _{n} = p _{n-1} $ , and such that $ X \rightarrow Y _{r} $ induces an isomorphism of homotopy groups in dimensions $ \leq r - 1 $ . Both towers can be used to calculate homotopy groups, cf. [a6] for some examples.

Closely related to killing homotopy groups are the ideas of obstruction theory [a5], cf. also Obstruction and Postnikov system.

To the three directions mentioned above there should now certainly be added the rational homotopy theory, developed by D. Sullivan based on "piecewise-linear differential forms" , cf. [a7], [a8] and also [a6].

References

[a1]	B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 30; 157–167
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 25; 437–444
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