In
mathematics, particularly
algebraic topology, cohomotopy sets are particular
contravariant functors from the
category of
pointed topological spaces and basepoint-preserving
continuous maps to the category of
sets and
functions. They are
dual to the
homotopy groups, but less studied.
Overview
The p-th cohomotopy set of a pointed
topological space X is defined by
![{\displaystyle \pi ^{p}(X)=[X,S^{p}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4625735b2dd3846da1c72394af0bd21c0f7a8ae0)
the set of pointed
homotopy classes of continuous mappings from
to the p-
sphere
.
[1]
For p = 1 this set has an
abelian group structure, and is called the Bruschlinsky group. Provided
is a
CW-complex, it is
isomorphic to the first
cohomology group
, since the
circle
is an
Eilenberg–MacLane space of type
.
A theorem of
Heinz Hopf states that if
is a
CW-complex of dimension at most p, then
is in
bijection with the p-th cohomology group
.
The set
also has a natural
group structure if
is a
suspension
, such as a sphere
for
.
If X is not homotopy equivalent to a CW-complex, then
might not be isomorphic to
. A counterexample is given by the
Warsaw circle, whose first cohomology group vanishes, but admits a map to
which is not homotopic to a constant map.
[2]
Properties
Some basic facts about cohomotopy sets, some more obvious than others:
for all p and q.
- For
and
, the group
is equal to
. (To prove this result,
Lev Pontryagin developed the concept of framed
cobordism.)
- If
has
for all x, then
, and the homotopy is smooth if f and g are.
- For
a
compact
smooth manifold,
is isomorphic to the set of homotopy classes of
smooth maps
; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.
- If
is an
-
manifold, then
for
.
- If
is an
-
manifold with boundary, the set
is
canonically in bijection with the set of cobordism classes of
codimension-p framed submanifolds of the
interior
.
- The
stable cohomotopy group of
is the
colimit
![{\displaystyle \pi _{s}^{p}(X)=\varinjlim _{k}{[\Sigma ^{k}X,S^{p+k}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a281b39e97bf8497d74c62cf8ced9abd2dde4532)
- which is an abelian group.
History
Cohomotopy sets were introduced by
Karol Borsuk in 1936.
[3] A systematic examination was given by
Edwin Spanier in 1949.
[4] The stable cohomotopy groups were defined by
Franklin P. Peterson in 1956.
[5]
References
-
^
"Cohomotopy_group",
Encyclopedia of Mathematics,
EMS Press, 2001 [1994]
-
^ "
The Polish Circle and some of its unusual properties". Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved November 16, 2023. See also the accompanying diagram "
Constructions on the Polish Circle"
-
^ K. Borsuk, Sur les groupes des classes de transformations continues, Comptes Rendue de Academie de Science. Paris 202 (1936), no. 1400-1403, 2
-
^ E. Spanier, Borsuk’s cohomotopy groups, Annals of Mathematics. Second Series 50 (1949), 203–245. MR 29170
https://doi.org/10.2307/1969362
https://www.jstor.org/stable/1969362
-
^ F.P. Peterson, Generalized cohomotopy groups, American Journal of Mathematics 78 (1956), 259–281. MR 0084136