Triad

Quadruples $ ( X; A, B, x _ {0} ) $, where $ X $ is a topological space and $ A $ and $ B $ are subspaces of it such that $ A \cup B = X $ and $ x _ {0} \in A \cap B $. The homotopy groups of triads, $ \pi _ {n} ( X; A, B, x _ {0} ) $, $ n \geq 3 $( for $ n = 2 $, it is just a set), have been introduced and are used in the proof of homotopy excision theorems. There is also an exact Mayer–Vietoris sequence connecting the homology groups of the spaces $ X $, $ A $, $ B $, $ A \cap B $( cf. Homology group).

Comments[edit]

For a triple $ ( X ; A, B) $ consisting of a space $ X $ and two subspaces $ A , B \subset X $, one defines the path space $ \Omega ( X; A, B) $ as the space of all paths in $ X $ starting in $ A $ and ending in $ B $,

$$ \Omega ( X; A, B) = \{ {p : [ 0, 1] \rightarrow X } : { p( 0) \in A , p( 1) \in B } \} . $$

If there is a distinguished point $ * $ in $ A \cap B $, the constant path at $ * $ is taken as a distinguished point of $ \Omega ( X; A, B) $( and is also denoted by $ * $).

The relative homotopy groups (cf. Homotopy group) $ \pi _ {n} ( X, A, * ) $, $ * \in A \subset X $, can also be defined as $ \pi _ {n-} 1 ( \Omega ( X; A, * ) , * ) $. Now let $ ( X; A, B, * ) $ be a triad. The homotopy groups of a triad are defined as the relative homotopy groups

$$ \pi _ {n} ( X; A, B , * ) = \ \pi _ {n-} 1 ( \Omega ( X; B, * ),\ \Omega ( A; A \cap B , * ), * ). $$

Using the long homotopy sequence of the triplet $ ( \Omega ( X; B, * ) , \Omega ( A; A \cap B, * ), * ) $ there results the (first) homotopy sequence of a triad

$$ {} \dots \rightarrow \pi _ {n+} 1 ( X; A, B, * ) \mathop \rightarrow \limits ^ \partial \pi _ {n} ( A, A \cap B, * ) \rightarrow $$

$$ \rightarrow \ \pi _ {n} ( X, B, * ) \rightarrow \pi _ {n} ( X; A, B, x _ {0} ) \mathop \rightarrow \limits ^ \partial \dots , $$

so that the triad homotopy groups measure the extend to which the homotopy excision homomorphisms

$$ \pi _ {n} ( A, A \cap B, * ) \rightarrow \pi _ {n} ( X, B, * ) $$

fail to be isomorphisms. The triad homotopy groups can also be defined as

$$ \pi _ {n} ( X; A, B, * ) = \pi _ {n-} 1 ( \Omega ( X; A, B), * ). $$

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