Fundamental class
The fundamental class of an -connected topological space
(that is, a topological space
such that
for
) is the element
of the group
that corresponds, under the isomorphism
that arises in the universal coefficient formula
![]() |
to the inverse of the Hurewicz homomorphism
(which is an isomorphism by the Hurewicz theorem (see Homotopy group)). If
is a CW-complex (a cellular space), then the fundamental class
is the same as the first obstruction to the construction of a section of the Serre fibration
, which lies in
, and also as the first obstruction to the construction of a homotopy of the identity mapping
to a constant mapping. In case the
-dimensional skeleton of
consists of a single point (in fact this assumption involves no loss of generality, since any
-dimensional CW-complex is homotopy equivalent to a CW-complex without cells of positive dimension less than
), the closure of each
-dimensional cell is an
-dimensional sphere, and so its characteristic mapping determines some element of the group
. Since these cells form a basis of the group
, it thus determines an
-dimensional cochain in
. This cochain is a cocycle and its cohomology class is also the fundamental class.
A fundamental class, or orientation class, of a connected oriented -dimensional manifold
without boundary (respectively, with boundary
) is a generator
of the group
(respectively, of
), which is a free cyclic group. If
can be triangulated, then the fundamental class is the homology class of the cycle that is the sum of all coherent oriented
-dimensional simplices of an arbitrary triangulation of it. For each
, the homomorphism
![]() |
where the -product is defined by the formula
![]() |
is an isomorphism, called Poincaré duality (if has boundary
, then
). One also speaks of the fundamental class for non-oriented (but connected) manifolds
(with boundary); in this case one means by it the unique element of
(respectively, of
) 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) |
Fundamental class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_class&oldid=47021