• Source: Homotopy excision theorem

In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let



(
X
;
A
,
B
)


{\displaystyle (X;A,B)}

be an excisive triad with



C
=
A
∩
B


{\displaystyle C=A\cap B}

nonempty, and suppose the pair



(
A
,
C
)


{\displaystyle (A,C)}

is (



m
−
1


{\displaystyle m-1}

)-connected,



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
and is surjective for



q
=
m
+
n
−
2


{\displaystyle q=m+n-2}

.
A geometric proof is given in a book by Tammo tom Dieck.
This result should also be seen as a consequence of the most general form of the Blakers–Massey theorem, which deals with the non-simply-connected case.
The most important consequence is the Freudenthal suspension theorem.




References






Bibliography


J. Peter May, A Concise Course in Algebraic Topology, Chicago University Press.

Kata Kunci Pencarian:

• Homotopy excision theorem
• Excision theorem
• Freudenthal suspension theorem
• Homotopy group
• Homotopy theory
• Hurewicz theorem
• List of algebraic topology topics
• Glossary of algebraic topology
• Blakers–Massey theorem
• List of theorems