Difference between revisions of "Triad"
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Revision as of 06:27, 27 May 2016
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
,
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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
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Using the long homotopy sequence of the triplet there results the (first) homotopy sequence of a triad
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so that the triad homotopy groups measure the extend to which the homotopy excision homomorphisms
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fail to be isomorphisms. The triad homotopy groups can also be defined as
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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 |
Triad. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triad&oldid=16998