This note presents a procedure of constructing a higher dimensional sphere map from a lower dimensional one and gives an explicit formula for smooth sphere map with a given degree. As an application a new proof of a generalized Poincare-Hopf theorem called Morse index formula is also presented.
The degree of (continuous) smooth map n n S S f → : is a well established topic appeared in many text books and a vast literature. Nonetheless, how to construct specific maps from n S to n S with given dimension and topological degree number seems seldom appear in the literature, although it is interesting and meaningful. In this note we present a procedure of constructing a higher dimensional sphere map from a lower dimensional one and give an explicit formula for sphere map with given degree number. Thus for each homotopic class of continuous map n n S S f → :
we can have an explicit representative map. As an application, we give a new proof of the generalized Poincare-Hopf theorem on indices of continuous vector fields, which was obtained by Morse [2], and rediscovered by Pugh [4] and Gottlleb [1]. For convenience, this generalized theorem is called Morse index formula following Grottleb [1].
In this section we prove a lemma that is essential to our construction of higher dimensional sphere map by lower dimensional sphere map and important for the new proof the Morse index formula.
) (x V be a continuous vector field defined on
to be the degree of the map
and suppose that the set
U , is a tangent vector field on n U , which also induces a lower dimensional sphere map
is defined to be the degree of the above map. Now we have the following observations. Lemma 2.1 Let ) (x V be a continuous vector field defined on
Without loss of generality we can assume that the first n components ) ( ),…, (
it is easy to see that this homotopy makes sense as long as r is sufficiently small.
To see what the index ) 0 , (W ind is, let us consider the degree of
, thus we prove the first case.
For case (ii), we can in the same manner define a new vector field ) ), ( ),…, ( ) (
By virtue of Lemma 2.1 it is easy to propose a procedure of constructing an explicit smooth map fromn dimensional sphere to itself with a given topological degree, as shown in the next section..
The purpose of this section is to present an explicit smooth map from
which has the origin as its only zero. With this vector field define the following map
) , , ,…, (
and define the map
We have the following statement
) , , ,…, (
and define the map
) , , ,…, (
We have the following statement
, then under the above assumptions, the so called Morse index formula can be stated as follows [1].
In this section we present a new proof based on the lemma given in Section 2. This new proof seems more understandable and is more general in the sense that it does not needs the assumption that zeros are non-degenerate, which is essential to the arguments by Pugh [4].
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