# 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

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

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.

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