Subject: I'm with Kuhn and Greenlees
Date: Thu, 9 May 2002 10:46:53 -0400
From: Tom Goodwillie <tomg@math.brown.edu>
To: Don Davis <dmd1@lehigh.edu>

I agree. In fact, I wish that "(n-1)-connected" was
called "n-connected" -- the first nontrivial dimension
is the key measure of how connected a space is. But it's too late
for that. Forget I mentioned it.

>I don't know if Adams' use of
>    X(m,n) (which would be better X[m,n])
>and X(m, ... , \infty ) (which would be better X[m,\infty) )
>
>in `On Chern characters and the structure of the unitary group'
>in 1961 might have given rise to the X<n> type notation.

In his "Student's Guide" to Algebraic Topology Adams made an effort to
fix this notation. Scattered through the book are various pieces
written by Adams himself, and (if memory serves) one of them is a
small fragment of the paper of his mentioned above,
included in the book for the express purpose of advertising his
notation for connective covers. I think we should all vow to follow
his lead, except for the change to pointy brackets.

Oh, and while I'm on my hobby horse, what about "connective" spectra?
I believe that "connective" means trivial homotopy groups below
dimension 0, in other words

"connective" means -1-connected,

while (at least for spaces, so maybe also for spectra)

"connected" means 0-connected.

How unfortunate. A person could get carried away and decide that
n-connected should be the same as (n-1)-connective.
On the other (also unfortunate) hand,
I have a hunch that there are some folks who add to the muddle by
using the term "connective spectrum" in slightly different ways.

Tom Goodwillie

P.S. (Still on the horse) the empty set is not connected.

P.P.S. I was startled recently when I heard a mathematician
(not a topologist) use the term 2-connected to mean that pi_n is
trivial for n>2. It sounded like his friends used it that way, too.