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.

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)