Let
and define
Claim
is contractible
Proof
Suppose
then
Define
by
by
by
and so on ...
applying the homotopy
in the time interval
,
in the interval
,
in
etc...
we get the desired contraction to the point
.
A different way to see this is via the cellular structure of
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you can always contract
along
like moving contracting the equator along the surface of the earth.