if f and g are two smooth functions from a smooth manifold M^n->S^p,
and we have the condition
that |f-g|<2 then f and g are smoothly homotopic. I think I
understand
that in this case f and g
are both members of a cohomotopy group, but the significance of <2
evades me. Clearly, if
|f-g|>2, then either one of both of the maps don't map onto the n-
sphere, if |f-g|=2, then f,g
map to antipodal points, but don't see right away why that scotches
their being homotopic.
Thanks in advance for any suggestions on the significance of the
inequality condition.
If |f-g|<2 you can pull each point f(x), x in M, along the shorter one of
the two great circle segments connecting it to g(x), and this gives a
homotopy from f to g.
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