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The fundamental class of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421601.png" />-connected topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421602.png" /> (that is, a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421603.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421604.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421605.png" />) is the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421606.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421607.png" /> that corresponds, under the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f0421608.png" /> that arises in the universal coefficient formula
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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/f/f042/f042160/f0421609.png" /></td> </tr></table>
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to the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216010.png" /> of the Hurewicz homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216011.png" /> (which is an isomorphism by the Hurewicz theorem (see [[Homotopy group|Homotopy group]])). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216012.png" /> is a [[CW-complex|CW-complex]] (a cellular space), then the fundamental class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216013.png" /> is the same as the first [[Obstruction|obstruction]] to the construction of a section of the [[Serre fibration|Serre fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216014.png" />, which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216015.png" />, and also as the first obstruction to the construction of a homotopy of the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216016.png" /> to a constant mapping. In case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216017.png" />-dimensional skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216018.png" /> consists of a single point (in fact this assumption involves no loss of generality, since any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216019.png" />-dimensional CW-complex is homotopy equivalent to a CW-complex without cells of positive dimension less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216020.png" />), the closure of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216021.png" />-dimensional cell is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216022.png" />-dimensional sphere, and so its characteristic mapping determines some element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216023.png" />. Since these cells form a basis of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216024.png" />, it thus determines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216025.png" />-dimensional [[Cochain|cochain]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216026.png" />. This cochain is a [[Cocycle|cocycle]] and its cohomology class is also the fundamental class.
+
The fundamental class of an ( n - 1) $-
 +
connected topological space  $  X $(
 +
that is, a topological space  $  X $
 +
such that  $  \pi _ {i} ( X) = 0 $
 +
for  $  i \leq  n - 1 $)
 +
is the element $  r _ {n} $
 +
of the group $  H  ^ {n} ( X;  \pi _ {n} ( X)) $
 +
that corresponds, under the isomorphism  $  H  ^ {n} ( X;  \pi ) \approx  \mathop{\rm Hom} ( H _ {n} ( X);  \pi ) $
 +
that arises in the universal coefficient formula
  
A fundamental class, or orientation class, of a connected oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216028.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216029.png" /> without boundary (respectively, with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216031.png" />) is a generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216032.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216033.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216034.png" />), which is a free cyclic group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216035.png" /> can be triangulated, then the fundamental class is the homology class of the cycle that is the sum of all coherent oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216036.png" />-dimensional simplices of an arbitrary triangulation of it. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216037.png" />, the homomorphism
+
$$
 +
0 \rightarrow \
 +
\mathop{\rm Ext} ( H _ {n - 1 }  ( X);  \pi )  \rightarrow \
 +
H  ^ {n} ( X;  \pi ) \rightarrow  \mathop{\rm Hom} ( H _ {n} ( X);  \pi ) \rightarrow  0,
 +
$$
  
<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/f/f042/f042160/f04216038.png" /></td> </tr></table>
+
to the inverse  $  h  ^ {-} 1 $
 +
of the Hurewicz homomorphism  $  h:  \pi _ {n} ( X) \rightarrow H _ {n} ( X) $(
 +
which is an isomorphism by the Hurewicz theorem (see [[Homotopy group|Homotopy group]])). If  $  X $
 +
is a [[CW-complex|CW-complex]] (a cellular space), then the fundamental class $  r _ {n} $
 +
is the same as the first [[Obstruction|obstruction]] to the construction of a section of the [[Serre fibration|Serre fibration]]  $  \Omega X \rightarrow EX \rightarrow X $,
 +
which lies in  $  H  ^ {n} A ( X, \pi _ {n - 1 }  ( \Omega X)) = H  ^ {n} ( X; \pi _ {n} ( X)) $,
 +
and also as the first obstruction to the construction of a homotopy of the identity mapping  $  \mathop{\rm id} : X \rightarrow X $
 +
to a constant mapping. In case the  $  ( n - 1) $-
 +
dimensional skeleton of  $  X $
 +
consists of a single point (in fact this assumption involves no loss of generality, since any  $  ( n - 1) $-
 +
dimensional CW-complex is homotopy equivalent to a CW-complex without cells of positive dimension less than  $  n $),
 +
the closure of each  $  n $-
 +
dimensional cell is an  $  n $-
 +
dimensional sphere, and so its characteristic mapping determines some element of the group  $  \pi _ {n} ( X) $.
 +
Since these cells form a basis of the group  $  C _ {n} ( X) $,
 +
it thus determines an  $  n $-
 +
dimensional [[Cochain|cochain]] in  $  C  ^ {n} ( X; \pi _ {n} ( X)) $.  
 +
This cochain is a [[Cocycle|cocycle]] and its cohomology class is also the fundamental class.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216039.png" />-product is defined by the formula
+
A fundamental class, or orientation class, of a connected oriented  $  n $-
 +
dimensional manifold  $  M $
 +
without boundary (respectively, with boundary  $  \partial  M $)
 +
is a generator  $  [ M] $
 +
of the group  $  H _ {n} ( M) $(
 +
respectively, of  $  H _ {n} ( M, \partial  M) $),
 +
which is a free cyclic group. If  $  M $
 +
can be triangulated, then the fundamental class is the homology class of the cycle that is the sum of all coherent oriented  $  n $-
 +
dimensional simplices of an arbitrary triangulation of it. For each  $  q $,
 +
the homomorphism
  
<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/f/f042/f042160/f04216040.png" /></td> </tr></table>
+
$$
 +
D _ {M} : \
 +
H  ^ {q} ( M)  \rightarrow \
 +
H _ {n - q }  ( M),\ \
 +
D _ {M} : \
 +
x  \rightarrow  x \cap [ M],
 +
$$
  
is an isomorphism, called [[Poincaré duality|Poincaré duality]] (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216041.png" /> has boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216043.png" />). One also speaks of the fundamental class for non-oriented (but connected) manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216044.png" /> (with boundary); in this case one means by it the unique element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216045.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042160/f04216046.png" />) different from zero. In this case there is also a Poincaré duality.
+
where the  $  \cap $-
 +
product is defined by the formula
 +
 
 +
$$
 +
x ( y \cap c)  = \
 +
( x \cup y) ( c) ,\ \
 +
\mathop{\rm dim}  x +
 +
\mathop{\rm dim}  y  = \
 +
\mathop{\rm dim}  c,
 +
$$
 +
 
 +
is an isomorphism, called [[Poincaré duality|Poincaré duality]] (if $  M $
 +
has boundary $  \partial  M $,  
 +
then $  D _ {M} : H  ^ {q} ( M) \rightarrow H _ {n - q }  ( M, \partial  M) $).  
 +
One also speaks of the fundamental class for non-oriented (but connected) manifolds $  M $(
 +
with boundary); in this case one means by it the unique element of $  H _ {n} ( M;  \mathbf Z _ {2} ) $(
 +
respectively, of $  H _ {n} ( M, \partial  M;  \mathbf Z _ {2} ) $)  
 +
different from zero. In this case there is also a Poincaré duality.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


The fundamental class of an $ ( n - 1) $- connected topological space $ X $( that is, a topological space $ X $ such that $ \pi _ {i} ( X) = 0 $ for $ i \leq n - 1 $) is the element $ r _ {n} $ of the group $ H ^ {n} ( X; \pi _ {n} ( X)) $ that corresponds, under the isomorphism $ H ^ {n} ( X; \pi ) \approx \mathop{\rm Hom} ( H _ {n} ( X); \pi ) $ that arises in the universal coefficient formula

$$ 0 \rightarrow \ \mathop{\rm Ext} ( H _ {n - 1 } ( X); \pi ) \rightarrow \ H ^ {n} ( X; \pi ) \rightarrow \mathop{\rm Hom} ( H _ {n} ( X); \pi ) \rightarrow 0, $$

to the inverse $ h ^ {-} 1 $ of the Hurewicz homomorphism $ h: \pi _ {n} ( X) \rightarrow H _ {n} ( X) $( which is an isomorphism by the Hurewicz theorem (see Homotopy group)). If $ X $ is a CW-complex (a cellular space), then the fundamental class $ r _ {n} $ is the same as the first obstruction to the construction of a section of the Serre fibration $ \Omega X \rightarrow EX \rightarrow X $, which lies in $ H ^ {n} A ( X, \pi _ {n - 1 } ( \Omega X)) = H ^ {n} ( X; \pi _ {n} ( X)) $, and also as the first obstruction to the construction of a homotopy of the identity mapping $ \mathop{\rm id} : X \rightarrow X $ to a constant mapping. In case the $ ( n - 1) $- dimensional skeleton of $ X $ consists of a single point (in fact this assumption involves no loss of generality, since any $ ( n - 1) $- dimensional CW-complex is homotopy equivalent to a CW-complex without cells of positive dimension less than $ n $), the closure of each $ n $- dimensional cell is an $ n $- dimensional sphere, and so its characteristic mapping determines some element of the group $ \pi _ {n} ( X) $. Since these cells form a basis of the group $ C _ {n} ( X) $, it thus determines an $ n $- dimensional cochain in $ C ^ {n} ( X; \pi _ {n} ( X)) $. This cochain is a cocycle and its cohomology class is also the fundamental class.

A fundamental class, or orientation class, of a connected oriented $ n $- dimensional manifold $ M $ without boundary (respectively, with boundary $ \partial M $) is a generator $ [ M] $ of the group $ H _ {n} ( M) $( respectively, of $ H _ {n} ( M, \partial M) $), which is a free cyclic group. If $ M $ can be triangulated, then the fundamental class is the homology class of the cycle that is the sum of all coherent oriented $ n $- dimensional simplices of an arbitrary triangulation of it. For each $ q $, the homomorphism

$$ D _ {M} : \ H ^ {q} ( M) \rightarrow \ H _ {n - q } ( M),\ \ D _ {M} : \ x \rightarrow x \cap [ M], $$

where the $ \cap $- product is defined by the formula

$$ x ( y \cap c) = \ ( x \cup y) ( c) ,\ \ \mathop{\rm dim} x + \mathop{\rm dim} y = \ \mathop{\rm dim} c, $$

is an isomorphism, called Poincaré duality (if $ M $ has boundary $ \partial M $, then $ D _ {M} : H ^ {q} ( M) \rightarrow H _ {n - q } ( M, \partial M) $). One also speaks of the fundamental class for non-oriented (but connected) manifolds $ M $( with boundary); in this case one means by it the unique element of $ H _ {n} ( M; \mathbf Z _ {2} ) $( respectively, of $ H _ {n} ( M, \partial M; \mathbf Z _ {2} ) $) different from zero. In this case there is also a Poincaré duality.

References

[1] D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[2] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[3] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[5] A. Dold, "Lectures on algebraic topology" , Springer (1980)
How to Cite This Entry:
Fundamental class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_class&oldid=47021
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article