2.1. Preliminaries
Let us recall some basic information on group actions. For a more detailed description, see [
21,
22]. Let
G be a group. Recall that a
G-set is a pair
, where
X is a set and
is the action of
G on
X, i.e., a map such that:
- (i)
for and ;
- (ii)
for , where is the group unit.
In the sequel, we write instead of , , , unless it leads to ambiguity.
Given G-sets X and Y, a map is G-equivariant if for any and . If the G-action on Y is trivial, then we say that f is G-invariant.
A subset of a G-set X is G-invariant if for all . The set is called the orbit through , and denotes the set of all orbits. Observe that if , then the set is G-invariant.
If G is a topological group, X is a topological space and a G-set, then X is a G-space, provided the action is (jointly) continuous. We say that a real (resp. complex) Banach space is a real (resp. complex) Banach representation of G if is a G-space, and for each , the map is linear and bounded. Throughout the whole paper, we assume that G is a compact Lie group.
Set-valued maps are the main object of our studies. Recall that given sets X and Y, a set-valued map from X into Y (written ) is a map that assigns to each the value , being a non-empty subset of Y. If X and Y are topological spaces and, for any closed (resp. open) set , the preimage is closed (resp. open), then we say that is upper (resp. lower) semicontinuous; is continuous if it is upper and lower semicontinuous simultaneously.
If Y is a metric space, then is lower semicontinuous if and only if for any , the function is upper semicontinuous (as a real function) or, equivalently, given and , .
A similar characterization of upper semicontinuity is not true, i.e., the lower semicontinuity of
does not imply in general that
is upper semicontinuous. However, if
has closed values, is locally compact, i.e., each point
has a neighbourhood
U such that
is compact, and
d is lower semicontinuous, then
is upper semicontinuous (with compact values). The graph
of an upper semicontinuous map
with closed values is closed;
is upper semicontinuous with compact values if and only if the projection
is perfect (recall that a continuous map
is perfect if it is closed and
is compact for any
). We say that a map
is compact if it is upper semicontinuous and the closure of the image
is compact. For other details on set-valued maps, see [
23] or [
18].
Let be metric spaces and . Recall the notion of graph approximations.
Definition 1. A continuous map is an ε-approximation of if for every x, there exists such that and .
One can formulate the above condition as follows:
for all
. It is also equivalent to the condition that the graph of
f is contained in the
-neighbourhood of the graph of
. It is well known that such approximations are a good tool for extending topological invariants, and they have been proven to exist for example for convex-valued u.s.c. maps; see [
5]. We are interested in equivariant versions of the results.
Definition 2. Let X and Y be G-sets. A set-valued map is G-equivariant (resp. G-invariant) if (resp. ) for all and .
Note that is G-equivariant if and only if for all and (or for all and ). Moreover, it is easy to see that is G-equivariant if and only if its graph is a G-invariant subset of with a natural action , , . Observe that if , where Y is a topological space, is G-equivariant, then so is its closure, i.e., the map given by , .
Let us collect simple examples of G-equivariant set-valued maps.
Example 1. (1) Let , where and is an open G-invariant subset of a Banach G-representation , be G-equivariant, i.e., for and . Consider a differential inclusion (under suitable assumptions assuring the existence of solutions): Consider the space of continuous maps from to with the G-action defined by , where for , and . If is a solution to the above problem, i.e., there is an integrable function such that , , then is also a solution to this problem with the initial condition . Therefore, the solution map that assigns to each initial value the set of all solutions is G-equivariant, whenever well defined.
(2) Let , where is a real Banach representation of G, be a convex function. For each , the subdifferential:is defined. Let f be G-invariant. Then, the map is G-equivariant. Clearly, is invariant, and if , then for all , . Hence:which gives the assertion. In a similar manner, one shows that the Clarke generalized gradient , where U is a G-invariant open in and is a G-invariant locally Lipschitz function, is G-equivariant. 2.2. Gradient approximations
In
, we shall use various analytic characterizations of graph-approximations. For
, we recall the notion of a support function
given by:
One easily checks that is finite and continuous if A is bounded. Moreover, .
We also define:
or equivalently:
Lemma 1. ([19], Lemma 2.2) Let , with closed convex values, , and . The following conditions are equivalent: - (i)
g is an ε-approximation of φ;
- (ii)
for every , there is such that: - (iii)
for every , there is such that:
Let
be an open subset, and consider a locally Lipschitz function
. A generalized directional Clarke derivative:
is defined for all
. Clarkés generalized gradient at
x is the set:
It is well known that the map
is an u.s.c. mapping with compact convex values (see [
17] for this and more of the properties of the generalized gradient). One observes also that
for
. Recall only the local u.s.c. property:
Definition 3. The map is upper semicontinuous at if: The following approximation result was proved in [
19]:
Theorem 1. Let be locally Lipschitz and a continuous function. Then, there exists a -map such that:
- (i)
for all ,
- (ii)
for all .
In other words, g is a uniform approximation of f, and is an -approximation of . We shall call such pairs Whitney approximations.
Definition 4. Suppose , where compact, is a locally Lipschitz function. If satisfies:
- 1.
for all (uniform ε-approximation),
- 2.
for all (ε-approximation (on the graph)),
for a continuous function , then we say that is the ε-WT-approximation (ε-Whitney-type-approximation) of .
Let us assume that V is an n-dimensional orthogonal representation of a compact Lie group G and is open and G-invariant. Assume that a locally Lipschitz function is G-invariant, i.e., for all . It is easy to verify that is then equivariant, i.e., as sets.
We now modify the proof from [
19] in order to obtain an equivariant version of Theorem 1.
Let us start from the remark that since
G acts orthogonally, there exists an invariant
-mollifier, i.e., a nonnegative invariant function
with the support
and such that:
We can take, e.g., a function given by:
and then, we normalize it:
Let
be an invariant
L-Lipschitz map with a compact support, and let
. We define the
-regularization of
f by the formula:
The following is a straightforward consequence of the definition.
Proposition 1. The regularization satisfies the following properties:
- (i)
is -smooth;
- (ii)
is G-invariant;
- (iii)
is a uniform -approximation of f;
- (iv)
.
The following technical lemma was proven in detail in [
19], Lemma 3.4.
Lemma 2. Let be Lipschitz functions of common rank L and with compact supports. For any , there exists sufficiently small with the property that for every , there exists such that:for each . A local approximation result is a consequence of the above.
Proposition 2. Let be G-invariant functions as in Lemma 2. Then, for any , there exists such that:
- (i)
for any , the regularization of is a uniform ε-approximation of , for each ;
- (ii)
for every , there exists such that for all and :
Proof. We put
, and let
. Then, Property (i) follows from Proposition 1. We prove (ii) by means of Property (iii) from Lemma 1. For this, we fix a point
x and
. There exists a
such that, for all
i, we have:
On the other hand, we have:
and by the use of Lemma 2, we find
such that:
Therefore, we have the inequality . This ends the proof because of the equality . □
Theorem 2. Let be a locally G-invariant map as in Theorem 1. Then, for every , there exists a -smooth G-invariant map such that is an ε-WT-approximation of .
Proof. Since
U is
G-invariant, there exists an increasing family of
G-invariant compact sets
such that:
For each k = 1, the set is also G-invariant. Therefore, the family is an open G-invariant covering of U. Let be a -smooth and G-invariant partition of unity subordinated to .
Thus, the functions are globally Lipschitz, G-invariant, and they can be considered as defined on .
We define a sequence of positive numbers:
Now, we can apply Proposition 2 for each
. We find
such that for any
invariant
-regularizations:
of
and
of
are uniform
-approximations. Moreover, for every
, there is
such that:
for all
. We can assume that
, and the sequence
is nonincreasing.
The desired map
is defined by the formula:
It is obviously well defined, since at most two terms are different from zero at given x. By definition, it is -smooth and G-invariant. We have to check the approximation conditions. For a fixed , there is a unique such that . If , then in some neighbourhood of x, and the claim is clear from the choice of .
If
, then in some neighbourhood of
x contained in
, we have
and
. Therefore:
By the properties of the generalized gradient, we obtain:
Because of Lemma 1, the last condition means that is a graph -approximation of . This ends the proof. □
Theorem 3. Let X be a compact G-invariant subset of V. Suppose that is locally Lipschitz and G-invariant. Then, for all , there exists a with and such that, for every two -approximations of , the pair is an -approximation of , where:is a linear homotopy. Proof. For all , is upper semicontinuous (1) at x, so we can apply Definition 3 with a given and obtain an appropriate, small .
We can assume that
for
, and
. Then, we have
. For all
, we observe that
implies
. The family
is an open covering of
X. We can choose a finite subcovering:
Let
be a Lebesgue number for the covering
, i.e.,:
and let
.
Fix
, and suppose that
are
-approximations of
. Then, we have:
and:
Then, from (2) and (3), we get:
From (2) and (3), we have:
so:
We can use the upper semicontinuity property of
in
:
for points
, which are close enough to
:
However,
is a convex set, and therefore:
Finally, from (2) and (4), we have:
This ends the proof. □
2.3. Equivariant Gradient Degree
In this section, we outline an axiomatic approach to the equivariant gradient degree. To this end, some preliminaries are in order.
Recall that the Euler ring
of a compact Lie group
G as an abelian group is equal to
, where
is the set of all conjugacy classes of normal subgroups of
G. The multiplicative structure of
is quite involved (see [
22] or [
4] for details on the ring structure).
Let V be an orthogonal G-representation. Denote by the space of G-invariant real -functions on V. Clearly, if , then the gradient map is G-equivariant. Let be an open, bounded and G-invariant subset of V and . A pair is said to be a G-gradient -admissible pair if for all , and there exists such that . Denote by the set of all G-gradient -admissible pairs, and put . In an obvious way, one can define a G-gradient -admissible homotopy between two G-gradient -admissible maps.
The notion of equivariant gradient degree we consider here was defined originally by K. Gȩba [
12]. We use an axiomatic approach here adapted from [
8].
Theorem 4. There exists a unique map , which assigns to every an element :satisfying the following properties ( are generators of the Euler ring, and are integer coefficients that may be nonzero only if the orbit type is present in the representation V): - 1.
(Existence) If , i.e., there is a nonzero coefficient , then such that and .
- 2.
(Additivity) Let be two disjoint, open, G-invariant subsets of Ω
and . Then: - 3.
(Homotopy) If is an admissible gradient G-equivariant homotopy, then: - 4.
(Normalization) Let be a special Ω
-Morse function such that and . Then,where “” stands for the total dimension of eigenspaces for negative eigenvalues of a (symmetric Hessian) matrix. - 5.
(Multiplicativity) For all ,where the multiplication “⋆” is taken in the Euler ring . - 6.
(Suspension) If W is an orthogonal G-representation and an open bounded invariant neighbourhood of , then: - 7.
(Hopf property) Assume that is the unit ball of an orthogonal G-representation V, and for , one has:Then, and are G-gradient -admissible homotopic.
Let us remark that in the case of the trivial action of
G, the above degree reduces to the classical Brouwer degree, as well as the usual non-gradient
G-equivariant degree (comp. [
8]). This follows from the celebrated A. Parusiński theorem (see [
24]), which says that every two gradient maps that are
-admissible homotopic are also gradient
-admissible homotopic. However, this is no longer true for equivariant maps.
Example 2. Take with the natural -action. Clearly, and are -equivariant. Moreover, is a -admissible -equivariant homotopy between Φ
and Ψ
. Clearly, this homotopy is not a gradient. Moreover, it follows from [11] that such a gradient -equivariant homotopy does not exist. Since the equivariant gradient degree satisfies the Hopf property with respect to gradient G-homotopies, as well as the ordinary equivariant degree with respect to equivariant homotopies, the gradient degree is more delicate homotopy invariant.
The ordinary equivariant degree for multivalued convex-valued u.s.c. maps was defined in [
7] by means of the equivariant version of the Cellina approximation theorem (see [
6]). The approximation results from the previous section enable us to develop the equivariant degree theory in quite an analogous way.
Let V be a finite dimensional orthogonal representation of a compact Lie group G, and let be an open, bounded G-invariant subset of V. Let be a locally Lipschitz G-invariant map such that its Clarkés generalized gradient is -admissible, i.e., for . Since the map is u.s.c., the image of the boundary is compact, and its distance from zero is positive. The same is true for a sufficiently fine neighbourhood of . Thus, there exists an such that for every , each -WT-approximation of is -admissible. Moreover, Theorem 3 assures that there exists a smaller such that all -approximations are -admissible homotopic by a linear homotopy. Take such a small .
Definition 5. We define the equivariant degree of Ω
-admissible pair as follows:where is a δ-WT-approximation (with g being G-invariant) of . The properties of the degree are consequences of Theorem 4. The first three of them are the most important for applications. Properties 1 and 2 are quite straightforward. The homotopy property needs some comment. Recall that the generalized gradient of a locally Lipschitz map is a multivalued u.s.c. map with convex compact values. Considering homotopies as continuous families of maps, we need here a stronger regularity than the u.s.c. property, which may admit quite a rapid change of values.
Definition 6. A gradient G-homotopy is a pair , where are locally Lipschitz G-invariant maps and the family of maps is continuous with respect to , as well as the family .
Observe that in the most popular case of linear homotopy
, the Clarke gradient with respect to the
x variable satisfies the condition:
In some cases, an equality holds, e.g., if at least one of the is of class , or if both of them are convex functions.
Theorem 5. Let the linear homotopy of locally Lipschitz maps be Ω
-admissible, and let the condition be satisfied for all x and . Then: Proof. The proof is similar to the proof of Theorem 3. For all , are upper semicontinuous (1) at x, so we can apply Definition 3 with a given and obtain an appropriate .
Let and . Then, .
For all
, we have that
implies
and
is an open covering of
X. We can choose a finite subcovering:
Let
be a Lebesgue number for the covering
, i.e.,:
and let
.
Fix
, and suppose that
are
-approximations of
, respectively. Then, we have:
and:
Then, from (5) and (6), we get:
From (5) and (6), we have:
so:
We can use the upper semicontinuity of
in
:
for points
:
This means that there exist
such that:
Therefore:
and
by our assumption. Thus:
Finally, from (5) and (7), we have:
This ends the proof. □
The above theorem is satisfactory in many applications. In particular, the Hopf property follows from this weak version of the homotopy property.
Theorem 6. Assume that is the unit ball of an orthogonal representation V and are Ω
-admissible pairs such that: Then, the pairs are Ω-admissible homotopic.
Proof. It is enough to observe that if we choose a
-WT-approximation
of
defining the equivariant degree, then the linear formula:
defines an
-admissible homotopy satisfying the assumptions of Theorem 5. Therefore, we apply this to both pairs. Now, we can apply the Hopf property for smooth maps
. The transitivity property of the homotopy relation for single-valued maps ends the proof. □
We are ready to prove the full homotopy property now.
Theorem 7. Let be an Ω
-admissible gradient G-homotopy. Then: Proof. First, we observe that because of the continuity of the homotopy with respect to , for each and every fixed t, there exist a and a Whitney approximation of , which is also an -Whitney approximation of for . Therefore, is linearly homotopic to , as well as to . Thus, by Theorem 5, the gradient degree is constant in the interval . Now, we consider the open covering of by the intervals (for , we put half-open intervals). We find a finite subcovering and apply the transitivity of the homotopy property to finish the proof. □