-dimensional and closed (or with boundary)
A finite simplicial complex with the following properties:
a) it is non-branching: Each -dimensional simplex is a face of precisely two (one or two, respectively) -dimensional simplices;
b) it is strongly connected: Any two -dimensional simplices can be joined by a "chain" of -dimensional simplices in which each pair of neighbouring simplices have a common -dimensional face;
c) it has dimensional homogeneity: Each simplex is a face of some -dimensional simplex.
If a certain triangulation of a topological space is a pseudo-manifold, then any of its triangulations is a pseudo-manifold. Therefore one can talk about the property of a topological space being (or not being) a pseudo-manifold.
Examples of pseudo-manifolds: triangulable, compact connected homology manifolds over (cf. Homology manifold); complex algebraic varieties (even with singularities); and Thom spaces (cf. Thom space) of vector bundles over triangulable compact manifolds. Intuitively a pseudo-manifold can be considered as a combinatorial realization of the general idea of a manifold with singularities, the latter forming a set of codimension two. The concepts of orientability, orientation and degree of a mapping make sense for pseudo-manifolds and moreover, within the combinatorial approach, pseudo-manifolds form the natural domain of definition for these concepts (especially as, formally, the definition of a pseudo-manifold is simpler than the definition of a combinatorial manifold). Cycles in a manifold can in a certain sense be realized by means of pseudo-manifolds (see Steenrod problem).
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Pseudo-manifold. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pseudo-manifold&oldid=24541