View source for Manifold crystallization

Coloured graphs (and crystallizations are a special class of them; cf. also [[Graph colouring|Graph colouring]]) constitute a nice combinatorial approach to the topology of piecewise-linear manifolds of any dimension. It is based on the facts that an edge-coloured graph provides precise instructions to construct a [[Polyhedron|polyhedron]], and that any piecewise-linear manifold (cf. [[Topology of manifolds|Topology of manifolds]]) arises in this way. The original concept is that of a contracted triangulation, due to M. Pezzana [[#References|[a9]]], which is a special kind of dissection of a [[Manifold|manifold]] yielding, in a natural way, a minimal atlas and a combinatorial representation of it via coloured graphs.

The graphs, considered in the theory, can have multiple edges but no loops. Given such a [[Graph|graph]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201001.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201003.png" /> denote the vertex set and the edge set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201004.png" />, respectively. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201006.png" />-coloured graph is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201007.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201008.png" /> is regular of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m1201009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010010.png" /> is a mapping (the edge-colouring) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010012.png" /> (the colour set) such that incident edges have different colours. The motivation for this definition is that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010013.png" />-coloured graph encodes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010014.png" />-dimensional complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010015.png" /> constructed as follows. Take an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010016.png" />-simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010017.png" /> for each vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010019.png" />, and label its vertices and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010020.png" />-faces by the colours of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010021.png" /> in such a way that vertices and opposite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010022.png" />-faces have the same label. Then each coloured edge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010023.png" /> indicates how to glue two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010024.png" />-simplexes along one of their common <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010025.png" />-faces (the colour says which). More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010027.png" /> are vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010028.png" /> joined by an edge coloured <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010029.png" />, then identify the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010030.png" />-faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010032.png" /> labelled by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010033.png" />, so that equally labelled vertices are identified together. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010034.png" /> is not, in general, a [[Simplicial complex|simplicial complex]] (two simplexes may meet in more than a single subsimplex), but it is a pseudo-complex, i.e. a ball complex in which each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010035.png" />-ball, considered with all its faces, is abstractly isomorphic to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010036.png" />-simplex.

The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010037.png" /> is called a crystallization of a closed connected piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010038.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010039.png" /> if the polyhedron underlying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010040.png" /> is piecewise-linearly homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010042.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010043.png" /> vertices (or, equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010044.png" /> is a contracted triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010045.png" />). The existence theorem of the theory says that any closed connected piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010046.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010047.png" /> can be represented by a crystallization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010048.png" /> in the sense made precise above [[#References|[a9]]]. This result can be extended to piecewise-linear manifolds with non-empty boundary and to piecewise-linear generalized (homology) manifolds by suitable modifications of the definition of crystallization. So, piecewise-linear manifolds can be studied through graph theory. Unfortunately, there are many different crystallizations representing the same manifold. However, two crystallizations represent piecewise-linear homeomorphic manifolds if and only if one can be transformed into the other by a finite sequence of elementary moves (i.e. cancelling and/or adding so-called dipoles) [[#References|[a7]]]. It follows that every topological invariant of a closed piecewise-linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010049.png" /> can be directly deduced from a crystallization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010050.png" /> via a graph-theoretical algorithm.

Below, a few such invariants are indicated; see [[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]] for more results and for further developments of crystallization theory.

==Orientability.==
A closed piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010051.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010052.png" /> is orientable if and only if a crystallization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010053.png" /> is bipartite (cf. also [[Graph, bipartite|Graph, bipartite]]), i.e. a graph whose vertex set can be partitioned into two sets in such a way that each edge joins a vertex of the first set to a vertex of the second set.

==Connected sums.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010055.png" /> be closed connected orientable piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010056.png" />-manifolds, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010058.png" /> be crystallizations of them. A crystallization for the connected sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010059.png" /> can be obtained as follows. Match arbitrarily the colours of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010060.png" /> with those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010061.png" />, and take away arbitrarily a vertex for either graph. Then past together the free edges with colours corresponding in the matching. This yields the requested crystallization since, by the disc theorem, the connected sum can be performed by hollowing out the two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010062.png" />-simplexes represented by the deleted vertices. The two permutation classes of matching correspond to an orientation-preserving, respectively an orientation-reversing, homeomorphism of the boundaries.

==Characterizations.==
An immediate characterization of coloured graphs representing piecewise-linear manifolds is provided by the following criterion. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010063.png" />-coloured graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010064.png" /> encodes a closed piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010065.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010066.png" /> if and only if any connected component of the partial subgraphs obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010067.png" /> by deleting all identically coloured edges, for each colour at a time, represents the standard piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010068.png" />-sphere. See [[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]] for other combinatorial characterizations of coloured graphs encoding low-dimensional manifolds.

==Homotopy and homology.==
A presentation of the [[Fundamental group|fundamental group]] of a closed connected piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010069.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010070.png" /> can be directly deduced from its crystallization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010071.png" /> as follows. Choose two colours <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010073.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010074.png" />, and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010075.png" />… <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010076.png" /> the connected components, but one, of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010077.png" />-subgraph obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010078.png" /> by deleting all edges coloured <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010079.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010080.png" /> (the missing component can be chosen arbitrarily). Of course, the connected components of the complementary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010081.png" />-subgraph are simple cycles with edges alternatively coloured <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010083.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010084.png" /> is a surface, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010085.png" /> be the unique cycle as above. If the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010086.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010087.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010088.png" /> these cycles, all but one arbitrarily chosen, and fix an orientation and a starting point for each of them. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010089.png" />, compose the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010090.png" /> on generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010091.png" /> by the following rules. Follow the chosen orientation starting from the chosen vertex, and write down consecutively every generator met with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010092.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010093.png" /> according to the colour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010094.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010095.png" /> of the edge leading to the generator. A presentation of the fundamental group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010096.png" /> has now generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010097.png" />, and relators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010098.png" />. A homology theory for coloured graphs was developed in [[#References|[a5]]], where one can find the graph-theoretical analogues to exact homology sequences, cohomology groups, product, duality, etc. and the corresponding topological meanings.

==Numerical invariants.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m12010099.png" /> be a crystallization of a closed connected piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100100.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100101.png" />. For each cyclic permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100103.png" />, there exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100104.png" />-cell imbedding (called regular; cf. also [[Graph imbedding|Graph imbedding]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100105.png" /> into a closed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100106.png" /> (which is orientable or non-orientable together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100107.png" />) such that its regions are bounded by simple cycles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100108.png" /> with edges alternatively coloured <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100110.png" /> (where the indices are taken modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100111.png" />). The regular genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100112.png" /> is defined as the smallest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100114.png" /> regularly imbeds into the closed (orientable or non-orientable) surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100115.png" />. The regular genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100116.png" /> is then the smallest of the regular genera of its crystallizations. A typical problem is to find relations between the regular genus of a manifold and the piecewise-linear structure of it. The topological classification of all closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100117.png" />-manifolds up to regular genus six can be found, for example, in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]. In particular, if the regular genus could be proved to be additive for connected sums in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100118.png" />, then this would imply the piecewise-linear generalized Poincaré conjecture in that dimension. Other numerical invariants of piecewise-linear manifolds arising from crystallizations, as for example many types of complexities, can be found in [[#References|[a1]]], [[#References|[a5]]].

==Geometric structure.==
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100119.png" />-coloured graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100120.png" /> is regular if its automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100121.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100122.png" /> (cf. also [[Graph automorphism|Graph automorphism]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100123.png" /> is locally regular if all the cycles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100124.png" />, with edges alternatively coloured <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100126.png" />, have the same number of vertices, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100127.png" />. If a locally regular graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100128.png" /> encodes a closed connected piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100129.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100130.png" />, then there is a regular graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100131.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100132.png" /> is isomorphic to a tessellation (cf. also [[Geometry of numbers|Geometry of numbers]]; [[Dirichlet tesselation|Dirichlet tesselation]]) by geometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100133.png" />-simplexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100134.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100135.png" /> is either the hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100136.png" />-space, the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100137.png" />-space or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100138.png" />-sphere, and there is a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100139.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100140.png" /> acting freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100141.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100142.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100143.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bracho,   L. Montejano,   "The combinatorics of coloured triangulations of manifolds"  ''Geom. Dedicata'' , '''22'''  (1987)  pp. 303–328</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Cavicchioli,   "A combinatorial characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100144.png" /> among closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100145.png" />-manifolds"  ''Proc. Amer. Math. Soc.'' , '''105'''  (1989)  pp. 1008–1014</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Cavicchioli,   "On the genus of smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100146.png" />-manifolds"  ''Trans. Amer. Math. Soc.'' , '''331'''  (1992)  pp. 203–214</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Cavicchioli,   M. Meschiari,   "On classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120100/m120100147.png" />-manifolds according to genus"  ''Cah. Topol. Géom. Diff. Cat.'' , '''34'''  (1993)  pp. 37–56</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Cavicchioli,   M. Meschiari,   "A homology theory for colored graphs"  ''Discrete Math.'' , '''137'''  (1995)  pp. 99–136</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Cavicchioli,   D. Repovš,   A.B. Skopenkov,   "Open problems on graphs arising from geometric topology"  ''Topol. Appl.'' , '''84'''  (1998)  pp. 207–226</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Ferri,   C. Gagliardi,   "Crystallization moves"  ''Pacific J. Math.'' , '''100'''  (1982)  pp. 85–103</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Ferri,   C. Gagliardi,   L. Grasselli,   "A graph-theoretical representation of PL-manifolds: A survey on crystallizations"  ''Aequat. Math.'' , '''31'''  (1986)  pp. 121–141</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Pezzana,   "Sulla struttura topologica delle varietà compatte"  ''Atti Sem. Mat. Fis. Univ. Modena'' , '''23'''  (1974)  pp. 269–277</TD></TR></table>