# Pseudo-manifold

$n$-dimensional and closed (or with boundary)

A finite simplicial complex with the following properties:

a) it is non-branching: Each $(n-1)$-dimensional simplex is a face of precisely two (one or two, respectively) $n$-dimensional simplices;

b) it is strongly connected: Any two $n$-dimensional simplices can be joined by a "chain" of $n$-dimensional simplices in which each pair of neighbouring simplices have a common $(n-1)$-dimensional face;

c) it has dimensional homogeneity: Each simplex is a face of some $n$-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 $\mathbf{Z}$; complex algebraic varieties (even with singularities); and Thom spaces 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).

#### References

 [1] H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) MR0575168 Zbl 0469.55001 [2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303