Erratum to "Fixed Points of Maps of a Nonaspherical Wedge"

Abstract

In the original paper, it was assumed that a selfmap of , the wedge of a real projective space and a circle , is homotopic to a map that takes to itself. An example is presented of a selfmap of that fails to have this property. However, all the results of the paper are correct for maps of the pair .

Let be the wedge of the real projective plane and the circle . As the example below demonstrates, the statement on page 3 of [1] "Given a map we may deform by a homotopy so that , its restriction to , maps to itself." is incorrect. If, instead of an arbitrary self-map of , we consider a map of pairs , the map can be put in the standard form defined on that page and then all the results of the paper are correct for such maps of pairs.

To describe the example, represent points of the unit -sphere by spherical coordinates where denotes the radius, the elevation and the azimuth. Let where is in or , if or , respectively. Let , where are the -spheres of radius one in with centers, in cartesian coordinates, at denotes the points for and the points for . Define in the following manner. For , let

(1)

in cartesian coordinates. For , set . Let be the poles and define by

(2)

Returning to cartesian coordinates, define by

(3)

We complete the definition of by setting for . Note that such that . We may embed in the universal covering space because is an infinite tree with a 2-sphere replacing each vertex in such a way that two edges are attached at each of two antipodal points. The embedding induces a monomorphism of homology. The map has been defined so that if are antipodal points of , then and therefore induces a map . If were homotopic to a map , then the homotopy would lift to cover by a map which sends to a single -sphere in . Therefore the image of would be either trivial or a single generator of . On the other hand, the image of in is nontrivial for three generators, so no such homotopy can exist. Therefore, if is a map whose restriction to is the map defined above, then it cannot be homotoped to a map that takes to itself.