Homotopy excision theorem

In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the homotopy excision theorem offers a substitute for the absence of <a href="/facts/Excision_theorem/GCH1wz2c">excision</a> in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>. More precisely, let 
 
 
 
 (
 X
 ;
 A
 ,
 B
 )
 
 
 {\displaystyle (X;A,B)}
 
 be an <a href="/facts/Excisive_triad/c74CTI6L">excisive triad</a> with 
 
 
 
 C
 =
 A
 ∩
 B
 
 
 {\displaystyle C=A\cap B}
 
 nonempty, and suppose the pair 
 
 
 
 (
 A
 ,
 C
 )
 
 
 {\displaystyle (A,C)}
 
 is <a href="/facts/N-connected/dbCgWgu3">(
 
 
 
 m
 −
 1
 
 
 {\displaystyle m-1}
 
)-connected</a>, 
 
 
 
 m
 ≥
 2
 
 
 {\displaystyle m\geq 2}
 
, and the pair 
 
 
 
 (
 B
 ,
 C
 )
 
 
 {\displaystyle (B,C)}
 
 is (
 
 
 
 n
 −
 1
 
 
 {\displaystyle n-1}
 
)-connected, 
 
 
 
 n
 ≥
 1
 
 
 {\displaystyle n\geq 1}
 
. Then the map induced by the inclusion 
 
 
 
 i
 :
 (
 A
 ,
 C
 )
 →
 (
 X
 ,
 B
 )
 
 
 {\displaystyle i\colon (A,C)\to (X,B)}
 
,

i
 
 ∗
 
 
 :
 
 π
 
 q
 
 
 (
 A
 ,
 C
 )
 →
 
 π
 
 q
 
 
 (
 X
 ,
 B
 )
 
 
 {\displaystyle i_{*}\colon \pi _{q}(A,C)\to \pi _{q}(X,B)}
 
,
is bijective for 
 
 
 
 q
 <
 m
 +
 n
 −
 2
 
 
 {\displaystyle q<m+n-2}
 
 and is surjective for 
 
 
 
 q
 =
 m
 +
 n
 −
 2
 
 
 {\displaystyle q=m+n-2}
 
. 
A geometric proof is given in a book by <a href="/facts/Tammo_tom_Dieck/b8mj7pGB">Tammo tom Dieck</a>.
This result should also be seen as a consequence of the most general form of the <a href="/facts/Blakers%25E2%2580%2593Massey_theorem/Xa4h9VcK">Blakers–Massey theorem</a>, which deals with the non-simply-connected case. 
The most important consequence is the <a href="/facts/Freudenthal_suspension_theorem/qGCDXhgl">Freudenthal suspension theorem</a>.

Homotopy excision theorem open-in-new

Homotopy excision theorem