Blakers–Massey theorem
Encyclopedia
In mathematics, the Blakers–Massey theorem, named after Albert Blakers and William S. Massey
, asserts that if X is the pushout
of
and f is m-connected and g is n-connected, then the map of pairs
induces an isomorphism
in relative homotopy group
s in degrees k ≤ (m + n − 1) and a surjection in the next degree.
Alternatively, it says that the pushout square above behaves like a homotopy pullback up to dimension m + n.
William S. Massey
William Schumacher Massey is an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including Algebraic Topology .William Massey was...
, asserts that if X is the pushout
Pushout
A pushout is a student that leaves his school before graduation, through the encouragement of the school. A student who leaves of his own accord , rather than through the action of the school, is considered a school dropout...
of
and f is m-connected and g is n-connected, then the map of pairs
induces an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
in relative homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s in degrees k ≤ (m + n − 1) and a surjection in the next degree.
Alternatively, it says that the pushout square above behaves like a homotopy pullback up to dimension m + n.