algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, a ''G''-fibration or principal fibration is a generalization of a principal ''G''-bundle, just as a fibration is a generalization of a fiber bundle. By definition, given a

topological monoid In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary operation is continuous. Every topologi ...

''G'', a ''G''-fibration is a fibration ''p'': ''P''→''B'' together with a continuous right

monoid action In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using th ...

''P'' × ''G'' → ''P'' such that *(1)

p(x g) = p(x)

for all ''x'' in ''P'' and ''g'' in ''G''. *(2) For each ''x'' in ''P'', the map

G \to p^(p(x)), g \mapsto xg

is a weak equivalence. A principal ''G''-bundle is a prototypical example of a ''G''-fibration. Another example is Moore's path space fibration: namely, let

P'X

be the space of paths of various length in a based space ''X''. Then the fibration

p: P'X \to X

that sends each path to its end-point is a ''G''-fibration with ''G'' the space of loops of various lengths in ''X''.

References

Algebraic topology Differential geometry Fiber bundles {{topology-stub