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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" />'' | + | {{TEX|done}} |
| + | ''$r$-dimensional Betti number $p^r$ of a complex $K$'' |
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− | 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" />: | + | 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 $S^n$: |
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− | <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>
| + | $$p^0=1,\quad p^1=\dotsb=p^{n-1}=0,\quad p^n=1;$$ |
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− | for the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598011.png" />: | + | for the projective plane $P^2(\mathbf R)$: |
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− | <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>
| + | $$p^0=1,\quad p^1=p^2=0;$$ |
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− | for the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015980/b01598013.png" />: | + | for the torus $T^2$: |
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− | <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>
| + | $$p^0=p^2=1,\quad p^1=2.$$ |
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− | 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 | + | For an $n$-dimensional complex $K^n$ the sum |
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− | <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>
| + | $$\sum_{k=0}^n(-1)^kp^k$$ |
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| 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]]]. |
Latest revision as of 13:05, 14 February 2020
$r$-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 $S^n$:
$$p^0=1,\quad p^1=\dotsb=p^{n-1}=0,\quad p^n=1;$$
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 |
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