Colloquium
3:00 p.m., Friday (December 3, 2004)
Math Annex 1100
Kee Lam
Department of Mathematics, UBC
The Kervaire Invariant: What we still don't know about the real projective plane
Let S^m={(x_0,x_1, ... ,x_m)\in \R^{m+1}x_0^2+x_1^2+ ... +x_m^2=1}
be the mdimensional sphere. Given m\ge n\ge 1, a primary concern of topologists
is to classify the (continuous) maps from S^m to S^n up to homotopy, where two maps
f, g are said to be homotopic iff f can be continuously deformed into g. In the
wellknown case when m=n=1, classification is attained by associating to f:S^1→ S^1
a winding number \omega (f)\in \Z, which is invariant when f undergoes continuous
deformation.
The Kervaire Invariant \alpha (f)\in \Z /2\Z of a map f:S^m→ S^n, definable via
framed cobordism for some values of m and n with m>n, is an invariant much more
subtle than the winding number. For the past 40 odd years, the search for maps f
with \alpha (f)\ne 0 continues to occupy a central stage in homotopy theory and
differential topology. In this talk I shall explain what the Kervaire invariant is,
and show how it can emerge from the study of certain questions intrinsic to the
geometry of the real projective plane. Some work in progress will be described,
while technical details will be kept to a minimum.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
