On the existence and classification ofisovariant maps
Ikumitsu NAGASAKI
Kyoto Prefectural University of Medicine (KPUM)

Sept. 2, 2010

Today’s talk(I) Existence or nonexistence results on isovariant maps,
in particular, Borsuk-Ulam type results.
(II) Classiﬁcation results on isovariant maps,
in particular, Hopf type results.

1

Isovariant maps
Let G be a compact Lie group. All maps are assumed to becontinuous.
Definition. A G-map f : X → Y between G-spaces iscalled G-isovariant if Gf (x) = Gx for all x ∈ X.
Example.
(1) If both X and Y are free G-spaces, then an arbitrary
G-map is a G-isovariant map.
(2) Suppose X is a G-space with nontrivial action andY is a G-space with Y G 6= ∅. In this case, a map
f : X → Y G ⊂ Y is equivariant, but not isovariant.

2

Isovariant homotopy classes
Definition. A G-homotopy F : X × I → Y is called aG-isovariant homotopy if F is G-isovariant.
Let [X, Y ]isov
G denote the set of isovariant homotopy classesof G-isovariant maps from X to Y .

As usual, [X, Y ]G denotes the set of G-homotopy classes ofG-maps from X to Y .

3Notation(1) Cn : a cyclic group of order n.
(2) Uk (= C), k ∈ Z: the unitary 1-dimensional representation of Cn on which a generator c ∈ Cn acts by√
kc · z = ξnz, where z ∈ Uk and ξn = exp(2π −1/n).
(3) SV : the unit sphere of a representation V of G, whichis called a representation sphere or linear G-sphere.

4An existence problem
Let G = Cpq , where p, q are distinct primes.
Set
U1r = U1 ⊕ · · · ⊕ U1 (r times)
andW = Up ⊕ Uq .
Note that G acts freely on SU1r , but not freely on SW .
In fact, the singular set (nonfree part) of SW :∪a>1CpCq
SW = SW
SW = SUp
SUq .

5

An existence problem
In equivariant case, as an application of equivariant
obstruction theory, one can see that, for any r ≥ 1, there
exists a Cpq -map
g : SU1r → S(Up ⊕ Uq ).
Question. What about a Cpq -isovariant map?
Does there exist a Cpq -isovariant map from SU1r to SW ?

6

The answer
If r = 1, then there is an isovariant map. For example, one
can deﬁne an isovariant map f0,0 : SU1 → SW by√
p qf0,0(z) = (z , z )/ 2.
(In fact, f0,0 is a G-embedding.)
When r ≥ 2, the answer is “No.”This is shown by a Borsuk-Ulam type theorem.

7

Borsuk Ulam type theorems
In transformation group theory, the classical Borsuk-Ulamtheorem is stated as follows.
Theorem 1. Let S m and S n be spheres with antipodal
C2-action. If there is a C2-map f : S m → S n, then theinequality m ≤ n holds.
Thus the Borsuk-Ulam theorem provides the nonexistence ofa C2-map. In fact, if m > n, then there is no C2-map from
S m to S n.

8

A generalization of the Borsuk-Ulam theorem
Many generalizations of the Borsuk-Ulam theorem are known.
The following is one of them.
Theorem 2 (N-Hara-Kawakami-Ushitaki, Biasi-de Mattos).
Let X be a free Cn-space and Y a Hausdorﬀ free Cn-space.
Suppose that there exists m ≥ 1 such thate q (X; Z/n) = 0 for 0 ≤ q ≤ m, andHHm+1(Y /Cn; Z/n) = 0.
Then there is no Cn-map from X to Y .
Here the homology is the singular homology.
9

A generalization of the Borsuk-Ulam theorem
This theorem deduces a well-known result below.
Corollary 3 (mod p Borsuk-Ulam theorem). Assume thatCp (p: prime) acts freely on X with H∗(X; Z/p) ∼=H∗(S m; Z/p) and on (Hausdorﬀ) Y with H∗(Y ; Z/p) ∼=H∗(S n; Z/p). If there is a Cp-map f : X → Y , then m ≤ n.
In other words, if m > n, then there is no Cp-map from Xto Y .

10Proof of the nonexistence
It suﬃces to show this when r = 2. Suppose
f : SU12 = S(U1 ⊕ U1) → S(Up ⊕ Uq ) = SW
is an isovariant map.
By restricting the action, we get a Cp-map f : SU12 →SW \ SW Cp
Since SW \ SW Cp ' S 1, SW \ SW Cp is a free Cp-homology
sphere of (homological) dimension 1.
By the mod p Borsuk-Ulam theorem, we have dim SU12 ≤ 1,
however this is a contradiction.

11

Homologically linear actions
The above example is generalized.
SetRG ={Z/|G| if dim G = 0,Z

if dim G > 0.

Definition. A smooth closed G-manifolds Σ is called
an RG-homologically linear G-sphere if for every (closed)
subgroup H, the H-ﬁxed point set ΣH is an RG-homology
sphere or the empty set; namely,H∗(ΣH ; RG) ∼= H∗(S m(H); RG), m(H) = dim ΣH .
For convenience, we set dim ΣH = −1 if ΣH is empty.
12

More general results
Then we haveTheorem 4 (Isovariant Borsuk-Ulam theorem). Let G bea solvable compact Lie group. Let Σ1 and Σ2 be RGhomologically linear G-spheres. If there is a G-isovariant
map f : Σ1 → Σ2, then the inequality
Gdim Σ1 − dim ΣG≤dim
Σ
−dim
Σ2
12holds.
Remark. Wasserman ﬁrst proved this result for representation spheres.

13

Nonsolvable case
Using a result of Oliver, we have
Proposition 5. If G is nonsolvable, then there exists asequencehnhn−1hn−2

h1· · · −−→ Σn −−−→ Σn−1 −−−→ · · · −→ Σ1such that• each Σn is a homologically linear G-sphere(in fact Σn can be taken to be a semilinear G-sphere),• each hn is a G-isovariant map,
• ΣGn =∅andlimn→∞ dim Σn = ∞.

14

Nonsolvable caseTake a G-embedding i : Σ1 ⊂ SW for some representation
W . Then an isovariant map fn : Σn → SW is deﬁned bycomposition.
Thus there is an integer n0 such thatdim Σn + 1 > dim SW − dim SW G for any n > n0.
This shows that the isovariant Borsuk-Ulam theorem does
not hold for a nonsolvable compact Lie group G.

15

RemarkHence, for RG-homologically linear actions, the isovariant
Borsuk-Ulam theorem holds if and only if G is solvable.
Remark. The problem whether the above Σn can be taken
to be a linear G-sphere is still open.

In equivariant case, the following is known.
Proposition 6 (Bartsch). Let G be a ﬁnite group. theBorsuk-Ulam theorem (in a weak sense) holds if and only if
G is of prime power order.

16

Corollary
Another result is obtained from the isovariant Borsuk-Ulamtheorem.
Corollary 7 (N-Ushitaki). Let G be a ﬁnite group and Σ
an RG-homology sphere with free G-action. Let SW be therepresentation sphere of a representation W of G. If there isa G-isovariant map f : Σ → SW , then the inequality
dim Σ + 1 ≤ dim SW − dim SW >1,where SW>1

=

∪

H
SW
(the singular set).
16=H≤G

Remark. This result still holds when G = S 1, Pin(2) ∼=NS 3 (S 1). However, in the case of G = S 3, it is unknown.
17

Existence results
The isovariant Borsuk-Ulam theorem provides nonexistence
results of isovariant maps.
we here discuss an existence problem under some conditions.
As is seen before, there is a Cpq -isovariant map
f0,0 : SU1 → S(Up ⊕ Uq ).
This is generalized as follows.

18

Existence results
Proposition 8. Let M be a free G-manifold. Let W be a
representation of G. Suppose
dim M/G + 1 ≤ dim SW − dim SW >1.
Then there exists a G-isovariant map from M to SW .

Outline of Proof.
Set SWfree = SW \ SW >1 and d = dim SW − dim SW >1.
Fact. SWfree is (d − 2)-connected.

19

Existence results
It suﬃces to construct a G-map from M to SWfree.
Fix a G-CW complex structure of M . One can inductively
construct a G-map as follows.
Suppose that a G-map fk : Xk → SWfree is constructed on
the k-skeleton Xk of M .
Let Xk+1 = Xk ∪φ G × Dk+1 ∪ · · · . Thenfk ◦ φ|∂Dk+1 : 1 × ∂Dk+1 → SWfree
is extended to fk+1| : Dk+1 → SWfree, since k ≤ d − 2 andSWfree is (d − 2)-connected. Hence fk equivariantly extends
to a G-map fk+1 : Xk+1 → SWfree.
20

Classification problem — An example
Next we discuss a classiﬁcation problem.
Recall the isovariant map

Let G = Cpq .

f0,0 : SU1 → SW = S(Up ⊕ Uq )√p qf0,0(z) = (z , z )/ 2.
One can ﬁnd other isovariant maps. Indeed, a map fα,β :SU1 → SW deﬁned by√p(1+αq) q(1+βp)fα,β (z) = (z,z)/ 2,fα,β is G-isovariant for (α, β) ∈ Z2.
Question. Do these maps represent diﬀerent isovariant
homotopy classes?
21Classification problem — An example
The answer is “Yes.” In fact,Proposition 9. If fα,β and fα0,β 0 are isovariantly homotopic,
then (α, β) = (α0, β 0).

In order to show this, we introduce the multidegree as anisovariant homotopy invariant.
If f is an isovariant map, then we obtain a G-map f : SU1 →SWfree.
Consider the induced homomorphismf∗ : H1(SU1) → H1(SWfree).
22Classification problem — An example
Lemma 10.
π1(SWfree) ∼= H1(SWfree) ∼= Z ⊕ Z.
Proof. SWfree = SW \(SUp ∪SUq ) ∼= (Up⊥ −0)×(Uq⊥ −0) 'SUq × SUp.
We deﬁne the multidegree mDeg (f ) of f bymDeg (f ) = f∗([SU1]) ∈ Z ⊕ Z.

23

Classification problem — An example
The multidegree of fα,β ismDeg fα,β = (q(1 + βp), p(1 + αq)).
This shows that if (α, β) 6= (α0, β 0), then mDeg fα,β 6=mDeg fα0,β 0 .
Hence the isovariant maps fα,β represent diﬀerent isovariant
homotopy classes.

24Classification problem — An example
In this case, the converse is true; in fact,
Proposition 11. Let f , g : SU1 → SW be isovariant
maps. If mDeg f = mDeg g, then f and g are isovariantly
homotopic.

Outline of Proof.
Set G = Cpq .
It suﬃces to construct a G-homotopy F : SU1 × I → SWfree
between f and g.

25

Classification problem — An example
Consider the commutative diagram:γG

1
2[SU1, SWfree]G −−→
H(SU/G,π)
=
Z
11∼
=


 ∗εyyπ∼=[SU1, SWfree] −→γ

H 1(SU1, π1) = Z2,where π1 = π1(SWfree) = Z2. The vertical map ε is theforgetful map and π : SU1 → SU1/G is the orbit map.

26

Classification problem — An example
γG

12[SU1, SWfree]G −−→
H(SU/G,π)=Z
11∼= ∗εyyπ∼=[SU1, SWfree] −→γ

H 1(SU1, π1) = Z2,

Fix a G-map g : SU1 → SWfree. The horizontal maps aredeﬁned byγG([f ]) = oG(f, g) and γ([f ]) = o(f, g),which are bijections as a consequence of the equivariant
obstruction theory.

27

Classification problem — An example
γG12[SU1, SWfree]G −−→
H(SU/G,π)=Z
11∼= ∗εyyπ∼=[SU1, SWfree] −→γ

H 1(SU1, π1) = Z2,One can see thatπ ∗ is multiplication by pqandπ ∗(oG(f, g)) = o(f, g).

28

Classification problem — An example
γG

12[SU1, SWfree]G −−→
H(SU/G,π)
=
Z
11
∼
=

 ∗εyyπ∼=[SU1, SWfree] −→γ

H 1(SU1, π1) = Z2,Hence π ∗ is injective, and the forgetful map ε is injective.
By calculation of the obstruction class, we haveγ([f ]) = o(f, g) = mDeg f − mDeg g.
Hence if mDeg f = mDeg g, then we have oG(f, g) = 0 and`so a G-map f g extends to a G-homotopy F .

29

A classification result
Furthermore it is seen thatmDeg f − mDeg g ∈ pqZ2.
Taking g = f0,0, we can deﬁne an injective map
D : [SU1, SWfree]G → Z ⊕ Z
by D[f ] = (mDeg f − mDeg f0,0)/pq.
Since D([fα,β ]) = (β, α), it follows that D is surjective.
Hence D is a bijection.

30

A classification result
Thus we have the following classiﬁcation result.
Proposition 12. There is a one-to-one correspondence
D : [SU1, SW ]isov
G → Z ⊕ Z.
In particular, the maps fα,β represent all isovariant homotopy
classes.

Using the notion of degree, H. Hopf showed thatdeg : [M, S n] → Z
is a bijection for an orientable closed n-manifold M . We callthis sort of result a Hopf type theorem.
31

A Hopf type theorem
The above example is generalized as follows.
We assume the following.
• G is a ﬁnite group.
• M is a connected, closed free G-manifold.
• SW is a unitary representation sphere of G.
• dim M + 1 = dim SW − dim SW >1.
Notation• A = { H ∈ Iso W | dim SW H = dim SW >1 }.
• A/G = { (H) | H ∈ A }.
32

A Hopf type theorem
Theorem 13 (Isovariant Hopf theorem). With the above
assumption(1) If M is orientable and the G-action on M is orientationpreserving, then there is a one-to-one correspondence⊕isov ∼
[M, SW ]G =Z.
(H)∈A/G

Every isovariant homotopy class is determined by themultidegree.

33

A Hopf type theorem(2) If M is non-orientable, then there is a one-to-one
correspondence⊕∼[M, SW ]isov
Z/2.
G =(H)∈A/G

If G is of odd order, then every isovariant homotopy
class is determined by the mod 2 multidegree.

34

Further results(1) In the case where M is orientable, if the G-action is
not orientation-preserving, then some Z/2 components
appear in [M, SW ]isov
G , and the multidegree does notdetermine the isovariant homotopy classes.
∼[M, SW ]isov
G =⊕Z⊕⊕

Z/2.

(2) In the case where M is non-orientable, if G is not ofodd order, then the mod 2 multidegree does not always
determine the isovariant homotopy classes.

35