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Difference between revisions of "Pseudo-manifold"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757202.png" />-dimensional and closed (or with boundary)''
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''$n$-dimensional and closed (or with boundary)''
  
A finite [[Simplicial complex|simplicial complex]] with the following properties:
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A finite [[simplicial complex]] with the following properties:
  
a) it is non-branching: Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757203.png" />-dimensional simplex is a face of precisely two (one or two, respectively) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757204.png" />-dimensional simplices;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757205.png" />-dimensional simplices can be joined by a "chain" of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757206.png" />-dimensional simplices in which each pair of neighbouring simplices have a common <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757207.png" />-dimensional face;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757208.png" />-dimensional simplex.
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c) it has dimensional homogeneity: Each simplex is a face of some $n$-dimensional simplex.
  
If a certain [[Triangulation|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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075720/p0757209.png" /> (cf. [[Homology manifold|Homology manifold]]); complex algebraic varieties (even with singularities); and Thom spaces (cf. [[Thom space|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|Steenrod problem]]).
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Examples of pseudo-manifolds: triangulable, compact connected [[homology manifold]]s over $\mathbf{Z}$; complex algebraic varieties (even with singularities); and [[Thom space]]s 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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) {{MR|0575168}} {{ZBL|0469.55001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) {{MR|0575168}} {{ZBL|0469.55001}} </TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.R. Munkres, "Elements of algebraic topology" , Addison-Wesley (1984) {{MR|0755006}} {{ZBL|0673.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) {{MR|0995842}} {{ZBL|0673.55002}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J.R. Munkres, "Elements of algebraic topology" , Addison-Wesley (1984) {{MR|0755006}} {{ZBL|0673.55001}} </TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) {{MR|0995842}} {{ZBL|0673.55002}} </TD></TR>
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</table>
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Latest revision as of 19:28, 13 October 2017

$n$-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


Comments

References

[a1] J.R. Munkres, "Elements of algebraic topology" , Addison-Wesley (1984) MR0755006 Zbl 0673.55001
[a2] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) MR0995842 Zbl 0673.55002
How to Cite This Entry:
Pseudo-manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-manifold&oldid=24541
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article