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
π
p
(
X
)
=
[
X
,
S
p
]
{\displaystyle \pi ^{p}(X)=[X,S^{p}]}
the set of pointed homotopy classes of continuous mappings from
X
{\displaystyle X}
to the p -sphere
S
p
{\displaystyle S^{p}}
.[ 1]
For p = 1 this set has an abelian group structure, and is called the Bruschlinsky group . Provided
X
{\displaystyle X}
is a CW-complex , it is isomorphic to the first cohomology group
H
1
(
X
)
{\displaystyle H^{1}(X)}
, since the circle
S
1
{\displaystyle S^{1}}
is an Eilenberg–MacLane space of type
K
(
Z
,
1
)
{\displaystyle K(\mathbb {Z} ,1)}
.
A theorem of Heinz Hopf states that if
X
{\displaystyle X}
is a CW-complex of dimension at most p , then
[
X
,
S
p
]
{\displaystyle [X,S^{p}]}
is in bijection with the p -th cohomology group
H
p
(
X
)
{\displaystyle H^{p}(X)}
.
The set
[
X
,
S
p
]
{\displaystyle [X,S^{p}]}
also has a natural group structure if
X
{\displaystyle X}
is a suspension
Σ
Y
{\displaystyle \Sigma Y}
, such as a sphere
S
q
{\displaystyle S^{q}}
for
q
≥
1
{\displaystyle q\geq 1}
.
If X is not homotopy equivalent to a CW-complex, then
H
1
(
X
)
{\displaystyle H^{1}(X)}
might not be isomorphic to
[
X
,
S
1
]
{\displaystyle [X,S^{1}]}
. A counterexample is given by the Warsaw circle , whose first cohomology group vanishes, but admits a map to
S
1
{\displaystyle S^{1}}
which is not homotopic to a constant map.[ 2]
Properties
Some basic facts about cohomotopy sets, some more obvious than others:
π
p
(
S
q
)
=
π
q
(
S
p
)
{\displaystyle \pi ^{p}(S^{q})=\pi _{q}(S^{p})}
for all p and q .
For
q
=
p
+
1
{\displaystyle q=p+1}
and
p
>
2
{\displaystyle p>2}
, the group
π
p
(
S
q
)
{\displaystyle \pi ^{p}(S^{q})}
is equal to
Z
2
{\displaystyle \mathbb {Z} _{2}}
. (To prove this result, Lev Pontryagin developed the concept of framed cobordism .)
If
f
,
g
:
X
→
S
p
{\displaystyle f,g\colon X\to S^{p}}
has
‖
f
(
x
)
−
g
(
x
)
‖
<
2
{\displaystyle \|f(x)-g(x)\|<2}
for all x , then
[
f
]
=
[
g
]
{\displaystyle [f]=[g]}
, and the homotopy is smooth if f and g are.
For
X
{\displaystyle X}
a compact smooth manifold ,
π
p
(
X
)
{\displaystyle \pi ^{p}(X)}
is isomorphic to the set of homotopy classes of smooth maps
X
→
S
p
{\displaystyle X\to S^{p}}
; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic.
If
X
{\displaystyle X}
is an
m
{\displaystyle m}
-manifold , then
π
p
(
X
)
=
0
{\displaystyle \pi ^{p}(X)=0}
for
p
>
m
{\displaystyle p>m}
.
If
X
{\displaystyle X}
is an
m
{\displaystyle m}
-manifold with boundary , the set
π
p
(
X
,
∂
X
)
{\displaystyle \pi ^{p}(X,\partial X)}
is canonically in bijection with the set of cobordism classes of codimension -p framed submanifolds of the interior
X
∖
∂
X
{\displaystyle X\setminus \partial X}
.
The stable cohomotopy group of
X
{\displaystyle X}
is the colimit
π
s
p
(
X
)
=
lim
→
k
[
Σ
k
X
,
S
p
+
k
]
{\displaystyle \pi _{s}^{p}(X)=\varinjlim _{k}{[\Sigma ^{k}X,S^{p+k}]}}
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