Whitehead multiplication
A multiplication in homotopy groups , defined by G.W. Whitehead . In
one takes a fixed decomposition into two cells
and
. Then the product of spheres
has a decomposition into cells
,
,
,
. Therefore the characteristic mapping
:
![]() |
factorizes as
![]() |
where is a bouquet of spheres. Now, take classes
and
, represented by mappings
and
. Then the Whitehead product
is given by the composition
![]() |
The following properties are satisfied by this product:
1) ;
2) if , then
;
3) if is
-simple, then
for
,
;
4) if for all
,
, then
is
-simple;
5) if ,
,
,
, then
![]() |
6) the element , where
is a generator, is equal to twice the generator of
;
7) the kernel of the epimorphism is generated by one element,
, where
is the canonical generator.
References
[1a] | G.W. Whitehead, "On products in homotopy groups" Ann. of Math. , 47 (1946) pp. 460–475 |
[1b] | G.W. Whitehead, "A generalization of the Hopf invariant" Ann. of Math. , 51 (1950) pp. 192–237 |
Comments
References
[a1] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) |
[a2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 419–420 |
[a3] | S.-T. Hu, "Homotopy theory" , Acad. Press (1959) pp. 138–139 |
Whitehead multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_multiplication&oldid=18732