Triad

Quadruples , where is a topological space and and are subspaces of it such that and . The homotopy groups of triads, , (for , 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 , , , (cf. Homology group).

Comments

For a triple consisting of a space and two subspaces , one defines the path space as the space of all paths in starting in and ending in ,

If there is a distinguished point in , the constant path at is taken as a distinguished point of (and is also denoted by ).

The relative homotopy groups (cf. Homotopy group) , , can also be defined as . Now let be a triad. The homotopy groups of a triad are defined as the relative homotopy groups

Using the long homotopy sequence of the triplet there results the (first) homotopy sequence of a triad

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

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

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