EXISTENCE AND CLASSIFICATION RESULTS
ON ISOVARIANT MAPS

Abstract. The notion of an isovariant map, i.e, an equivariant map preserving
the isotropy subgroups, plays an important role in equivariant topology. In thisarticle, we shall illustrate existence and classiﬁcation results on isovariant maps
and provide some examples.

Ikumitsu NAGASAKI (京都府立医科大学 数学教室・長崎 生光)
Department of Mathematics
Kyoto Prefectural University of Medicine

1. Isovariant maps
Let G be a compact Lie group. All maps considered are continuous. We ﬁrst
recall the notion of an isovariant map which was introduced by Palais [12].
Deﬁnition.
(1) A G-map f : X → Y is called G-isovariant if f preserves theisotropy subgroups, i.e., Gf (x) = Gx for all x ∈ X.
(2) A G-homotopy F : X × I → Y between isovariant maps is called a Gisovariant homotopy if F is G-isovariant.
We denote by [X, Y ]isov
the isovariant homotopy set, i.e., the set of isovariant
Ghomotopy classes of G-isovariant maps from X to Y . As usual [X, Y ]G denotes theset of equivariant homotopy classes of G-equivariant maps from X to Y .
Example 1.1.
(1) If both X and Y are free G-spaces, then an arbitrary G-map
f : X → Y is isovariant.
(2) An injective G-map f is G-isovariant.
(3) Suppose that X is a G-space with nontrivial action and Y has a G-ﬁxed
point. Then a constant map f : X → Y G ⊂ Y is equivariant, but notisovariant.
Let Cn be the cyclic group of order n. Let denote by Uk (= C) the unitary
representation of Cn on which a generator c ∈ Cn acts by c · z = ξ k z, where z ∈ Uk√and ξ = exp(2π −1/n).
2000 Mathematics Subject Classiﬁcation. 57S17, 55M20, 55M35.
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For simplicity, we consider the case of G = Cpq , where p, q are distinct primes.
Set
U1r = U1 ⊕ · · · ⊕ U1 (r times)
andW = Up ⊕ Uq .
Consider the unit spheres SU1r and SW . Then G acts freely on SU1r . On the other
hand G does not act freely on SW and the singular set (i.e., nonfree part)aSW >1 = SUp
SUq
is a Hopf link.
Question. Does there exist a Cpq -isovariant map from SU1r to SW ?
In equivariant case, a Cpq -map does exist for any r ≥ 1 as a result from equivariant obstruction theory, see [15]. What about a Cpq -isovariant map?
If r = 1, then there exists an isovariant map. For example, the equivariant map
f0,0 : SU1 → SW can be deﬁned by√
f0,0 (z) = (z p , z q )/ 2,
and it is isovariant. In fact, f0,0 is an injective G-map.
If r ≥ 2, the answer is “No.” This is proved by a Borsuk-Ulam type theorem.
Many generalizations of the Borsuk-Ulam theorem are known. The following result
is one of them.
Theorem 1.2 (mod p Borsuk-Ulam theorem). Assume that Cp (p: prime) acts
freely on mod p homology spheres Σ1 and Σ2 . If there is a Cp -map f : Σ1 → Σ2 ,then dim Σ1 ≤ dim Σ2 . In other words, if dim Σ1 > dim Σ2 , then there is no Cp -map
from Σ1 to Σ2 .
The nonexistence of an isovariant map is proved as follows. Suppose that f :SU1r → SW is a Cpq -isovariant map. By restricting the action, we regard f as aCp -isovariant map. Since f (SU1r ) ⊂ SW \ SW Cp , we obtain a Cp -map f : SU1r →SW \ SW Cp . Since SW \ SW Cp is Cp -homotopy equivalent to SUq , we obtain aCp -map g : SU1r → SUq between free Cp -spheres. The mod p Borsuk-Ulam theorem
says that
2r − 1 = dim SU1r ≤ dim SUq = 1,
however this is a contradiction when r ≥ 2.
The above example is generalized as follows. Set(Z/|G| if dim G = 0,
RG =Z
if dim G > 0.
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Deﬁnition. A smooth closed G-manifold 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.
Theorem 1.3 (Isovariant Borsuk-Ulam theorem [7]). Let G be a solvable compact
Lie group. Let Σ1 and Σ2 be RG -homologically linear G-spheres. If there is a
G-isovariant map f : Σ1 → Σ2 , then the inequality
Gdim Σ1 − dim ΣG
1 ≤ dim Σ2 − dim Σ2holds.
Remark. Wasserman ﬁrst proved this type of result for representation spaces.
If G is nonsolvable, the above result does not hold. In fact, using a result of Oliver[11], we can construct a counterexample for an arbitrary nonsolvable compact Liegroup. Thus we obtain thatTheorem 1.4 ([7]). For homologically linear actions, the isovariant Borsuk-Ulamtheorem holds if and only if G is solvable.
Remark. For (ordinary) linear actions, it is known [16] that there are nonsolvablegroups for which the isovariant Borsuk-Ulam theorem holds.
Remark. A similar result is already known in equivariant case. In fact, Bartsch[1] showed that for linear actions, the Borsuk-Ulam theorem holds (in a certain
equivariant sense) if and only if G is of prime power order.
Using the isovariant Borsuk-Ulam theorem, we obtain a generalization of thenonexistence result mentioned before.
Corollary 1.5 ([8]). Let G be a ﬁnite group and Σ an RG -homology sphere withfree G-action. Let SW be a representation sphere of G. If
dim Σ ≥ dim SW − dim SW >1 ,then there is no G-isovariant map form Σ to SW . Here SW >1 =(the singular set of SW ).
As seen before, there is a Cpq -isovariant map
f0,0 : SU1 → S(Up ⊕ Uq ).
This is generalized as follows.
3S

1̸=H≤G SW

HProposition 1.6 ([8]). Let G be a ﬁnite group. Let M be a free G-manifold andSW a representation sphere of G. Suppose
dim M < dim SW − dim SW >1 .
Then there exists a G-isovariant map from M to SW .
This follows from the fact that the complement of the singular set SW \ SW >1
is (d − 2)-connected, where d = dim SW − dim SW >1 .
2. Classification problem
Next we discuss the classiﬁcation problem on isovariant maps. Historically Hopf
showed that the degree of maps provides a bijection deg : [M, S n ] → Z, where M
is a connected, oriented, closed n-manifold M . We call this kind of result a Hopf
type theorem. Several equivariant versions of Hopf’s theorem (i.e., equivariant Hopf
type theorems) have been studied by many researchers, e.g., Rubinsztein [13], tom
Dieck [2], Tornehave [14], Laitinen [6], Kushkuley-Balanov [5], Ferrario [3], etc.
We would like to consider an isovariant version. Let G = Cpq again. For each(α, β) ∈ Z ⊕ Z, we can deﬁne a G-isovariant map
fα,β : SU1 → S(Up ⊕ Uq )by√fα,β (z) = (z p(1+αq) , z q(1+βp) )/ 2.

(The map f0,0 is the same one deﬁned previously.) Then we obtain
Proposition 2.1. If fα,β and fα′ ,β ′ are isovariantly homotopic if and only if(α, β) = (α′ , β ′ ).
To show this, we introduce the multidegree as an isovariant homotopy invariant.
Set SW = S(Up ⊕ Uq ) and SWfree = SW \ SW >1 . If f is an isovariant map, thenwe obtain a G-map
f : SU1 → SWfree
Consider the induced homomorphismf∗ : H1 (SU1 ) → H1 (SWfree ).
Since SWfree is the complement of a Hopf link in S 3 , which is homotopy equivalent
to S 1 × S 1 , we obtain that
π1 (SWfree ) ∼= H1 (SWfree ) ∼= Z ⊕ Z.
We deﬁne the multidegree mDeg (f ) of f bymDeg (f ) = f∗ ([SU1 ]) ∈ Z ⊕ Z.
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Lemma 2.2. The multidegree of fα,β ismDeg fα,β = (q(1 + βp), p(1 + αq)).
This shows that if (α, β) ̸= (α′ , β ′ ), then mDeg fα,β ̸= mDeg fα′ ,β ′ . Thus theisovariant maps fα,β mutually represent diﬀerent isovariant homotopy classes.
The converse is proved by equivariant obstruction theory. There is a commutative diagram:γG→ H 1 (SU1 /G, π1 ) = Z ⊕ Z[SU1 , SWfree ]G −−∼−
= ∗εyyπ∼=[SU1 , SWfree ] −−−→ H 1 (SU1 , π1 ) = Z ⊕ Z,γwhere π1 = π1 (SWfree ) ∼= Z ⊕ Z. The vertical map ε is the forgetful map. Thehorizontal maps are deﬁned byγG ([f ]) = oG (f, f0,0 )andγ([f ]) = o(f, f0,0 )by using the (equivariant) obstruction to construct a G-homotopy between f andf0,0 . The homomorphism π ∗ is the induced homomorphism of the orbit map π,which is multiplication by |G| = pq. In particular π ∗ is injective. By calculation of
the obstruction class, we obtain
Proposition 2.3.
π ∗ (oG (f, f0,0 )) = o(f, f0,0 ) = mDeg f − mDeg f0,0 ∈ pq(Z ⊕ Z).
We deﬁne a map
D : [SU1 , SWfree ]G → Z ⊕ Zby
D([f ]) = (mDeg f − mDeg f0,0 )/pq.
Since D([fα,β ]) = (β, α), D is surjective. Furthermore D is injective. In fact,
if D([f ]) = D([g]), then it follows from the above proposition that oG (f, f0,0 ) =oG (g, f0,0 ). This implies oG (f, g) = 0, and hence f and g are G-homotopic. Thus
D is a bijection. Since [SU1 , SW ]isov= [SU1 , SWfree ]G , we obtain the following
Gclassiﬁcation result.
Proposition 2.4. There is a bijection
D : [SU1 , SW ]isov
Cpq → Z ⊕ Z.
In particular, the maps fα,β represent all isovariant homotopy classes.
The above proposition is generalized as follows. We assume the following.
• G is a ﬁnite group.
• M is a connected, closed free G-manifold.
5• SW is a unitary representation sphere of G.
• dim M = dim SW − dim SW >1 − 1.
We setA = { H ∈ Iso W | dim SW H = dim SW >1 },andA/G = { (H) | H ∈ A }.
We have
Theorem 2.5 (Orientable case [9]). If M is orientable and the G-action on M isorientation-preserving, then there is a one-to-one correspondenceMZ.
[M, SW ]isov ∼=G(H)∈A/G

Every isovariant homotopy class is determined by the multidegree.
Remark. If the G-action is not orientation-preserving, the multidegree does notalways determine isovariant homotopy classes. Indeed, Z/2 components appear in[M, SW ]isov
G , see [9] for the detail.
Theorem 2.6 (Nonorientable case [10]). If M is nonorientable, then there is aone-to-one correspondenceM[M, SW ]isov ∼Z/2.
=G(H)∈A/GIf G is of odd order, then every isovariant homotopy class is determined by the mod2 multidegree.
Remark. If G is not of odd order, then the mod 2 multidegree does not always
determine isovariant homotopy classes.
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[9] I. Nagasaki and F. Ushitaki, A Hopf type classiﬁcation theorem for isovariant maps fromfree G-manifolds to representation spheres, to appear.
6[10] I. Nagasaki and F. Ushitaki, Isovariant maps from nonorientable free G-manifolds to representation spheres, in preparation.
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[12] R. S. Palais, Classiﬁcation of G-spaces, Mem. Am. Math. Soc. 36 (1960).
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Department of Mathematics, Kyoto Prefectural University of Medicine, 13
Nishitakatsukasa-cho, Taishogun Kita-ku, Kyoto 603-8334 Japan
E-mail address: nagasaki@koto.kpu-m.ac.jp

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