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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).

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

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

#### References

 [a1] S.-T. Hu, "Homotopy theory" , Acad. Press (1955) pp. Chapt. V, §10 [a2] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 88 [a3] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. §6.17
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