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Difference between revisions of "Betti number"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159802.png" />-dimensional Betti number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159803.png" /> of a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159804.png" />''
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{{TEX|done}}
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''$r$-dimensional Betti number $p^r$ of a complex $K$''
  
The rank of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159805.png" />-dimensional [[Betti group|Betti group]] with integral coefficients. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159806.png" /> the Betti number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159807.png" /> is a topological invariant of the polyhedron which realizes the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159808.png" />, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b0159809.png" />:
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The rank of the $r$-dimensional [[Betti group|Betti group]] with integral coefficients. For each $r$ the Betti number $p^r$ is a topological invariant of the polyhedron which realizes the complex $K$, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere $S^n$:
  
<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/b/b015/b015980/b01598010.png" /></td> </tr></table>
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$$p^0=1,\quad p^1=\ldots=p^{n-1}=0,\quad p^n=1;$$
  
for the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598011.png" />:
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for the projective plane $P^2(\mathbf R)$:
  
<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/b/b015/b015980/b01598012.png" /></td> </tr></table>
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$$p^0=1,\quad p^1=p^2=0;$$
  
for the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598013.png" />:
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for the torus $T^2$:
  
<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/b/b015/b015980/b01598014.png" /></td> </tr></table>
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$$p^0=p^2=1,\quad p^1=2.$$
  
For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598015.png" />-dimensional complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598016.png" /> the sum
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For an $n$-dimensional complex $K^n$ the sum
  
<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/b/b015/b015980/b01598017.png" /></td> </tr></table>
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$$\sum_{k=0}^n(-1)^kp^k$$
  
 
is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[1]]].
 
is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[1]]].

Revision as of 14:37, 1 August 2014

$r$-dimensional Betti number $p^r$ of a complex $K$

The rank of the $r$-dimensional Betti group with integral coefficients. For each $r$ the Betti number $p^r$ is a topological invariant of the polyhedron which realizes the complex $K$, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere $S^n$:

$$p^0=1,\quad p^1=\ldots=p^{n-1}=0,\quad p^n=1;$$

for the projective plane $P^2(\mathbf R)$:

$$p^0=1,\quad p^1=p^2=0;$$

for the torus $T^2$:

$$p^0=p^2=1,\quad p^1=2.$$

For an $n$-dimensional complex $K^n$ the sum

$$\sum_{k=0}^n(-1)^kp^k$$

is equal to its Euler characteristic. Betti numbers were introduced by E. Betti [1].

References

[1] E. Betti, Ann. Mat. Pura Appl. , 4 (1871) pp. 140–158


Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
How to Cite This Entry:
Betti number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Betti_number&oldid=16078
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article