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Erratum to "Fixed Points of Maps of a Nonaspherical Wedge"
Fixed Point Theory and Applications volume 2010, Article number: 820265 (2010)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F820265/MediaObjects/13663_2010_Article_1349_Equ1_HTML.gif)
in cartesian coordinates. For , set
. Let
be the poles and define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F820265/MediaObjects/13663_2010_Article_1349_Equ2_HTML.gif)
Returning to cartesian coordinates, define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F820265/MediaObjects/13663_2010_Article_1349_Equ3_HTML.gif)
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.
References
Kim SW, Brown RF, Ericksen A, Khamsemanan N, Merrill K: Fixed points of maps of a nonaspherical wedge. Fixed Point Theory and Applications 2009, 2099:-18.
Acknowledgment
The authors thank Francis Bonahon and Geoffrey Mess for their help with the example.
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The online version of the original article can be found at 10.1155/2009/531037
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Kim, S., Brown, R., Ericksen, A. et al. Erratum to "Fixed Points of Maps of a Nonaspherical Wedge". Fixed Point Theory Appl 2010, 820265 (2010). https://doi.org/10.1155/2010/820265
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DOI: https://doi.org/10.1155/2010/820265