Fundamental cocycle

of a cellular space $ X $ whose $ ( n - 1) $- skeleton $ X ^ {n - 1 } $ is a point $ x _ {0} $

The cochain in $ C ^ {n} ( X, \pi _ {n} ( X)) $ whose value on the cell $ e _ {i} ^ {n} $ is the element of $ \pi _ {n} ( X, x _ {0} ) $ corresponding to the closure $ \overline{ {e _ {i} ^ {n} }}\; $. The cohomology class of the fundamental cocycle is the fundamental class of $ X $.