Re: definition of homotopy group

On May 28, 9:00 pm, lrudo...@xxxxxxxxx (Lee Rudolph) wrote:
charles.cado...@xxxxxxxxxxxxxx writes (using more TeX than is
strictly consistent with legibility):



Take the following definition for the k-th homotopy group \pi_k(X,q)
of a space X at q \in X:
fix p \in S^k and call I the closed unit interval; then \pi_k(X,q)
is the set of homotopy classes of continuous pointed maps
f: (S^k, p) \times I \to (X, q)
where a homotopy between any two such maps, say f and g, is defined to
be a
continuous map
h: S^k \times I \to X
such that for any x \in X
h(x,0)=f(x)
h(x,1)=g(x)
h(p,0)=h(p,1)=q

The difference from the standard definition is in the last condition,
which usually is
h(p,t)=q for any t \in I.
Am I right in thinking that this is no real difference? After all, if
I
let the base point move around as t runs in I, the image of the
sphere doesn't "open up". Can it be proved or disproved it is the
same?

You have defined a set, but you have not yet defined a binary operation
on it under which it becomes a group. In "the standard definition" of
pi_k(X,q), the definition of the group operation uses the condition
that you have replaced with an apparently weaker form of itself, and
I believe you will find it a challenge to define a reasonable group
operation without using the strong original form of that condition.
I further believe that if you succeed in meeting that challenge, the
group you will have defined will not "be" pi_k(X,q) in any natural
sense (of course, in many cases cardinality considerations would
allow many unnatural definitions). Your set is a quotient set of
pi_k(X,q), but not (in general) a quotient group; for instance, for
k = 1, I believe your set is the set of conjugacy classes of
pi_1(X,q), otherwise known as "free homotopy classes of maps
of S^1 into X"--by construction, you've undone the "pointedness"
you began by assuming. Of course if pi_1(X,q) is abelian this
is the same as pi_1(X,q) itself (and the same as H_1(X,Z), for
X connected and suitably locally nice). More generally, if the
natural action of pi_1(X,q) on the set of pointed maps is trivial,
then your definition does give the correct quotient set, and you
can add a consistent definition of the binary operation that gives
the correct group structure on that set; but for non-trivial actions
everything goes kablooey.

Or so I think while I'm groggy with sleep. Having recently disgraced
myself with my silly remark about homotopy types of algebraic varieties,
I shouldn't expect to be taken seriously on homotopy (or much else),
at least not when I'm groggy.

Lee Rudolph

Yeah sorry for my useless TeX and my post infested by mistakes and
imprecisions (pointed out very nicely by Mariano Suárez-Alvarez on
sci.math)
Thank you for your comment, it does make sense to me. It seems I am
groggier than you.
CC

.

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