Re: definition of homotopy group

charles.cadogan@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

.

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