View source for Homotopy type

A class of homotopy-equivalent topological spaces. Two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479402.png" /> are said to be mutually-inverse homotopy equivalences if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479404.png" />. If only the first condition is met, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479405.png" /> is said to be a homotopy monomorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479406.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479407.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479408.png" /> dominates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h0479409.png" />. If there exists a homotopy equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794013.png" /> is said to be a weak homotopy equivalence if it induces an isomorphism of the homotopy groups in all dimensions (cf. [[Homotopy group|Homotopy group]]). Correspondingly, two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794015.png" /> are said to be weakly homotopy equivalent if there exists either a weak homotopy equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794016.png" /> or a weak homotopy equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794017.png" />. 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|CW-complex]]). This theorem is based on the facts that: 1) a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794018.png" /> is a homotopy equivalence if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794019.png" /> is a [[Deformation retract|deformation retract]] of the cylinder (cf. [[Mapping cylinder|Mapping cylinder]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794021.png" />; 2) a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794022.png" /> is a weak homotopy equivalence if and only if the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794023.png" /> of the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794024.png" /> is homotopy representative (cf. [[Representative subspace|Representative subspace]]); and 3) a subdivision of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794025.png" />-complex is representative if and only if it is a deformation retract.

Thus, the problem of homotopy types on the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794026.png" />-complexes is equivalent with the problem of weak homotopy types. On the other hand, any space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794027.png" /> is weakly homotopy equivalent to the geometric realization of its singular [[Simplicial set|simplicial set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794028.png" />. For this reason, the problem of weak homotopy types need be considered, without restricting generality, only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794029.png" />-complexes.

Two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794030.png" /> are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794032.png" />-homotopic if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794033.png" />-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794034.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794035.png" /> and any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794036.png" /> the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794038.png" /> are homotopic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794039.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794040.png" />-complex, this holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794041.png" />. Spaces that are equivalent with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794042.png" />-homotopy are called spaces of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794044.png" />-homotopy type. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794045.png" />-complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794047.png" /> are said to be complexes of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794049.png" />-type (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794050.png" />) if their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794051.png" />-th skeletons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794053.png" /> have the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794054.png" />-homotopy type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794056.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794057.png" />. This remains true even for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794058.png" />, if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794060.png" />-type is understood to be the homotopy type. In other words, the concept of being of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794061.png" />-type is homotopy invariant. The importance of the concept of being of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794062.png" />-type for the problem of homotopy types is due to the fact that two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794063.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794064.png" />-complexes are homotopy equivalent if and only if they have the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794065.png" />-type.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794066.png" /> be an arbitrary space (for the sake of simplicity — a totally-connected space). A simplicial subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794067.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794068.png" /> is said to be minimal if it contains all the singular simplices that are mappings into some fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794069.png" />, and if for any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794070.png" /> with all faces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794071.png" /> there exists a unique simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794072.png" /> that is homotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794073.png" /> (with respect to the boundary of the standard simplex). Minimal subsets exist and, up to an isomorphism, are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794074.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794075.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794076.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794077.png" />-dimensional [[Standard simplex|standard simplex]], considered as a simplicial decomposition (with respect to its standard triangulation), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794078.png" /> be its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794079.png" />-dimensional cochain group over an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794080.png" /> (more exactly, the group of normalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794081.png" />-dimensional cochains of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794082.png" />, cf. [[Cochain|Cochain]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794083.png" /> be a simplicial set in which the simplices of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794084.png" /> are cochains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794085.png" />, and the boundary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794086.png" /> and degeneracies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794087.png" /> are cochain mappings induced by the standard simplicial mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794089.png" /> (the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794090.png" /> "makes free" the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794091.png" />-th vertex, while the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794092.png" /> "glues" the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794093.png" />-th and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794094.png" />-th vertices). The simplices that are cocycles form a certain simplicial subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794095.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794096.png" />. The coboundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794097.png" /> defines a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794098.png" />, whose kernel is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h04794099.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940100.png" /> is a [[Fibration|fibration]] (in the sense of Kan) with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940101.png" />. In addition, the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940102.png" /> is an object of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940103.png" /> in the category of simplicial sets (with respect to homotopy groups in the sense of Kan, see [[Eilenberg–MacLane space|Eilenberg–MacLane space]]), while the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940104.png" /> is homotopy trivial (homotopy equivalent to a "point" ). Thus, the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940105.png" /> is a simplicial analogue of the [[Serre fibration|Serre fibration]] of paths over a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940106.png" />.

The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940107.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940108.png" /> is also meaningful for any (not necessarily Abelian) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940109.png" />. The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940110.png" /> thus obtained is the standard simplicial [[Resolution|resolution]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940111.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940112.png" /> be a multiplicative group of operators on the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940113.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940114.png" /> be an arbitrary simplicial set over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940115.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940116.png" /> be a cocycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940117.png" />. In the group of cocycles of this set over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940118.png" /> there is defined a coboundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940119.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940120.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940121.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940122.png" />-dimensional simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940123.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940124.png" /> be its characteristic mapping (cf. [[Simplicial set|Simplicial set]]). This defines a cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940125.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940126.png" />. Let the coboundary operator with respect to this cocycle be denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940127.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940128.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940129.png" />-dimensional cocycle of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940130.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940131.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940132.png" />. If in the direct product of the simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940134.png" /> one considers the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940135.png" /> consisting of all possible pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940138.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940139.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940140.png" /> is a simplicial subset. The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940141.png" /> defines a surjective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940142.png" />, which is a fibration (in the sense of Kan). This fibration will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940143.png" />. If the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940144.png" /> is trivial, then this fibration is identical with the fibration induced by the simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940145.png" /> which corresponds to the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940146.png" /> in the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940147.png" />. The fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940148.png" /> over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940149.png" />-dimensional skeleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940150.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940151.png" /> has a [[Section|section]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940153.png" /> is an [[Obstruction|obstruction]] to an extension of this section to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940154.png" />. After identification of the simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940156.png" /> it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940157.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940158.png" />.

Consider the following sequence of fibrations of simplicial sets:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940159.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Its initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940160.png" /> is the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940161.png" /> constructed from the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940162.png" />. By definition, one-dimensional simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940163.png" /> are in natural bijective correspondence with the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940164.png" />. Putting such a simplex into correspondence with the corresponding element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940165.png" /> yields some one-dimensional cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940166.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940167.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940168.png" />. Let the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940169.png" /> be defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940170.png" /> by the inductive formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940171.png" />. The sequence (1) is called a homotopy resolvent, or a [[Postnikov system|Postnikov system]] (the original name was: natural system), if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940172.png" /> the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940173.png" /> is a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940174.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940175.png" /> is a cocycle of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940176.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940177.png" /> over some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940178.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940179.png" /> with respect to the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940180.png" /> (and the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940181.png" /> is the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940182.png" />). This sequence is called the resolution of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940183.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940184.png" /> a simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940185.png" /> is given which is an isomorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940186.png" /> and which is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940187.png" />. The resolution uniquely determines the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940188.png" /> up to an isomorphism. On the other hand, the resolution itself is uniquely determined by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940189.png" /> and the cocycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940190.png" />. For this reason, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940191.png" />, consisting of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940192.png" /> and cocycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940193.png" />, may also be called a resolution.

Not all simplicial sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940194.png" /> have a resolution. According to the basic theorem in the theory of homotopy resolutions, a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940195.png" /> has a resolution if and only if it is isomorphic to the minimal simplicial set of a certain topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940196.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940197.png" />.

The resolution of the minimal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940198.png" /> is constructed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940199.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940200.png" />-dimensional simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940201.png" />. This simplex represents a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940202.png" /> that maps all vertices of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940203.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940204.png" />. For this reason it defines some element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940205.png" /> on any one-dimensional boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940206.png" />. Thus appears on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940207.png" /> a one-dimensional cocycle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940208.png" />, i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940209.png" />-dimensional simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940210.png" />; it will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940211.png" />. In this way one obtains a certain (simplicial) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940212.png" />. This mapping is an isomorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940213.png" /> and an epimorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940214.png" />. The next step is induction: Let the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940215.png" /> and the simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940216.png" />, which is an isomorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940217.png" /> and an epimorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940218.png" />, have been already constructed for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940219.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940220.png" /> has a right inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940221.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940222.png" /> be an obstruction to extension of this mapping to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940223.png" />. The obstruction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940224.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940225.png" />-dimensional cocycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940226.png" /> over the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940227.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940228.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940229.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940230.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940231.png" /> the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940232.png" /> is compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940233.png" />, and for this reason the difference element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940234.png" /> is defined (cf. [[Difference-element-in-K-theory|Difference element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940235.png" />-theory]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940236.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940237.png" />-dimensional simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940238.png" />. On each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940239.png" />-dimensional boundary of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940240.png" /> it defines a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940241.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940242.png" />. Assigning to this face the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940243.png" /> yields a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940244.png" />-dimensional cochain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940245.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940246.png" />, i.e. some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940247.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940248.png" /> of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940249.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940250.png" /> belongs to the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940251.png" />. To finish the induction one has to observe that the constructed mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940252.png" /> is a simplicial isomorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940253.png" /> and an epimorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940254.png" />.

The resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940255.png" /> is not uniquely constructed from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940256.png" />: there is freedom in the selection of the inverse mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940257.png" />. The simplest way of describing this non-uniqueness is to consider resolutions in the sense of (1). In fact, two such resolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940258.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940259.png" /> result from the same minimal simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940260.png" /> if and only if they are isomorphic as sequences of mappings, i.e. if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940261.png" /> there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940262.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940263.png" />. In order to describe such an isomorphism in terms of the resolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940264.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940265.png" /> it should be noted that the existence of an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940266.png" /> is equivalent to the existence of an isomorphism of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940267.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940268.png" />. Furthermore, for the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940269.png" /> there is a subsequent isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940270.png" /> if and only if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940271.png" />-isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940272.png" /> (see [[Operator homomorphism|Operator homomorphism]]) and a cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940273.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940274.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940275.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940276.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2prm)</td></tr></table>

Resolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940277.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940278.png" /> arise from the same simplicial set if and only if there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940279.png" /> that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940280.png" />-isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940281.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940282.png" /> relation (2) holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940283.png" /> is the isomorphism subsequently defined by (2prm) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940284.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940285.png" /> it is the isomorphism induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940286.png" />. In this case the resolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940287.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940288.png" /> are called isomorphic. The resolution of the symplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940289.png" /> is called the homotopy resolution of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940290.png" />. Summarizing, two spaces are weakly homotopy equivalent if and only if their homotopy resolutions are isomorphic; in particular, two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940291.png" />-complexes are homotopy equivalent if and only if their homotopy resolutions are isomorphic.

If (2) is satisfied only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940292.png" />, then the isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940293.png" /> only exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940294.png" />. In this situation one says that the given resolutions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940296.png" />-isomorphic. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940297.png" />-complexes are of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940298.png" />-type if and only if their homotopy resolutions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940299.png" />-isomorphic.

The given solution to the problem of homotopy types (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940300.png" />-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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940301.png" /> which form part of their solutions. If the space being studied is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940302.png" />-connected, then its resolution starts in fact with the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940303.png" />. If the solution of a given problem can be formulated in terms of the first non-trivial group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940304.png" /> only, this problem is called a problem of order zero (e.g. the Hopf–Whitney problem on the classification of mappings of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940305.png" />-dimensional polyhedron into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940306.png" />-connected space). If the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940308.png" /> and the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940309.png" /> are used, the problem is called a problem of order one (e.g. the problem of classifying mappings from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940310.png" />-dimensional polyhedron into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940311.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940312.png" />-connected space the homology class of the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940313.png" /> can be effectively computed; it has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940314.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940315.png" /> is the fundamental class of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940316.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940317.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940318.png" /> is the [[Steenrod operation|Steenrod operation]] corresponding to the natural [[Pairing|pairing]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940319.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940320.png" /> some invariant of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940321.png" /> is known as the [[Pontryagin square|Pontryagin square]]. For problems of higher orders it is necessary to effectively compute the next cocycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940322.png" />. Each of these cocycles is obtained from the fundamental class by some [[Cohomology operation|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,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940323.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

in which each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940324.png" /> has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940325.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940326.png" />, is called a resolution. This sequence is called the resolution of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940327.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940328.png" /> a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940329.png" /> is defined that induces an isomorphism of the homotopy groups in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940330.png" />, and that is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940331.png" />. This resolution is uniquely determined (up to an isomorphism, cf. [[Sequence category|Sequence category]]) by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940332.png" /> and the characteristic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940333.png" /> of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940334.png" />. The resolution exists for any totally-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940335.png" /> (such as the geometric realization of the "algebraic" resolution (1)) and determines this space up to weak homotopy equivalence. A fibre of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940336.png" /> is a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940337.png" /> and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940338.png" /> is homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940339.png" />-simple (cf. [[Homotopy group|Homotopy group]]), e.g. simply-connected, this fibration is induced by the [[Serre fibration|Serre fibration]] of paths over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940340.png" /> using a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940341.png" /> representing the cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940342.png" /> (cf. [[Eilenberg–MacLane space|Eilenberg–MacLane space]]; [[Representable functor|Representable functor]]).

If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940343.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940344.png" />-connected, then its resolution actually starts with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940345.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940346.png" /> it is convenient to regard, alongside with the "absolute" resolution (3), the resolution modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940347.png" />, the definition of which differs from the definition of (3) only in that the homotopy groups are replaced by their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940348.png" />-components. If the resolutions modulo any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940349.png" /> have been found for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940350.png" />, 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|Spectral sequence]]). For some spaces the computation of the resolution has been advanced sufficiently far.

For example, for the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940351.png" /> (for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940352.png" /> so that the stability condition is fulfilled) sufficiently many terms of its resolution modulo 2 are known. It is sufficient to describe the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940353.png" /> (i.e. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940354.png" />-components of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940355.png" />) and the cohomology classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940356.png" />. The first groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940357.png" /> have the following form:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940358.png" /></td> <td colname="2" style="background-color:white;" colspan="1">0</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">2</td> <td colname="5" style="background-color:white;" colspan="1">3</td> <td colname="6" style="background-color:white;" colspan="1">4</td> <td colname="7" style="background-color:white;" colspan="1">5</td> <td colname="8" style="background-color:white;" colspan="1">6</td> <td colname="9" style="background-color:white;" colspan="1">7</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940359.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940360.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940361.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940362.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940363.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940364.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940365.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940366.png" /></td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940367.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940368.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940369.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940370.png" /> is the fundamental class. The next class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940371.png" /> has the property that on the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940372.png" /> of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940373.png" /> it cuts out the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940374.png" />, and it is uniquely determined by it. Similarly, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940375.png" /> is uniquely characterized by the fact that, by reducing modulo 2, it transforms to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940376.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940377.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940378.png" /> vanish and the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940379.png" /> is uniquely characterized by the fact that on the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940380.png" /> of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940381.png" /> it cuts out the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940382.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940383.png" /> is characterized by the fact that, by reducing modulo 2, it transforms to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940384.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Brown, "Finite computability of Postnikov complexes" ''Ann. of Math. (2)'' , '''65''' (1957) pp. 1–20 {{MR|0083733}} {{ZBL|0077.16804}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.E. Mosher, M.C. Tangora, "Cohomology operations and their applications in homotopy theory" , Harper &amp; Row (1968) pp. Chapt. 13 {{MR|0394627}} {{ZBL|}} </TD></TR></table>



====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 [[#References|[a4]]] one should distinguish three directions in homotopy theory: suspension theory, which separates stable phenomena from those which are not (cf. [[Suspension|Suspension]], and also [[Cohomology operation|Cohomology operation]] and [[Homotopy group|Homotopy group]]), explicit geometric constructions such as Whitehead products (cf. [[Whitehead product|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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940385.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940386.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940387.png" />-connected, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940388.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940389.png" />, the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940390.png" /> has as fibre the Eilenberg–MacLane space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940391.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940392.png" />), and where above dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940393.png" /> the homotopy groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940394.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940395.png" /> agree.

In contrast, the Postnikov approximation (Postnikov tower) of a connected CW-complex is a sequence of fibrations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940396.png" /></td> </tr></table>

with the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940397.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940398.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940399.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940400.png" /> induces an isomorphism of homotopy groups in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047940/h047940401.png" />. Both towers can be used to calculate homotopy groups, cf. [[#References|[a6]]] for some examples.

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 30; 157–167 {{MR|0402714}} {{ZBL|0322.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 25; 437–444 {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 23; 415–455 {{MR|0516508}} {{ZBL|0406.55001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.F. Adams, "Algebraic topology: a students guide" , Cambridge Univ. Press (1972) {{MR|0445484}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H.J. Baues, "Obstruction theory" , Springer (1977) {{MR|0467748}} {{ZBL|0361.55017}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) pp. Chapt. I, Sect. 5 {{MR|0658304}} {{ZBL|0496.55001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Ph.A. Griffiths, J.W. Morgan, "Rational homotopy theory and differential forms" , Birkhäuser (1981) {{MR|0641551}} {{ZBL|0474.55001}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Sullivan, "Infinitesimal calculations in topology" ''Publ. Math. IHES'' , '''47''' (1977) pp. 269–331</TD></TR></table>