Subject: Re: names of htpy classes
Date: Thu, 26 Sep 2002 21:09:53 +0100
From: "Brian Sanderson" <bjs@maths.warwick.ac.uk>

> From: Ezra Getzler <getzler@math.northwestern.edu>

..stuff deleted..

> Japanese has a complex system of counters: long thin objects are counted
>
> by the suffix -hon (ippon, nihon, ...), small round objects by -ko, ...
> I wonder what the counter for homotopy classes is.
>
> Ezra Getzler

Similarly the Chinese have a prefix. I wonder what that is for homotopy
classes.

Brian Sanderson
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Subject: Re: names of htpy classes
Date: Thu, 26 Sep 2002 16:14:47 -0400 (EDT)
From: Tom Goodwillie <tomg@math.brown.edu>

So it's just a coincidence that an upper-case eta looks like
H for Hopf.
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Subject: Re: names of htpy classes
Date: Thu, 26 Sep 2002 21:36:05 -0500
From: Bill Richter <richter@math.northwestern.edu>

   From: Ezra Getzler <getzler@math.northwestern.edu>

   In partial support of Rognes's interesting conjecture on homotopy
   groups

   of spheres, four in Japanese is yon (epsilon). Actually, it would
   be better were epsilon replaced by upsilon. The next one would then
   be gamma (five=go).

That's interesting, Ezra, as first really new element after sigma is
epsilon, and I think the next really new element is mu.  Dunno what
upsilon or gamma is in Toda's book.

Epsilon is in the 8 stem, born on S^3 with Hopf invariant nu^2.  The 8
stem also has nu-bar = {nu, eta, nu}, born on S^6 with Hopf inv nu,
and eta sigma = epsilon + nu-bar.  Maybe nu-bar reminds Toda too much
of nu' = {eta, 2iota, eta} to fork off a new greek letter.  And indeed
in the Lambda algebra, nu-bar = \5 \3 = Sq^0(nu') = Sq^0(\2 \1).

I say the next really new element after epsilon is mu in the 8 stem,
born on S^3 with Hopf inv sigma''', Toda's name for the unstable
element which stably is 8 sigma.

That's why I use the phrase `first really new element', since indeed
you have to work to get sigma', sigma'', and sigma''', but Toda is
good enough at this to just go with primes.  Similarly we have nu' in
the 3 stem, born on S^3 with Hopf inv eta, which is the first example
of a Toda bracket, nu' = {eta, 2iota, eta}, due earlier to Barratt,
who also proved that H(nu') = eta because of the facts

 eta . 2iota = [iota_2, iota_2] on S^2,  and  2iota . eta = 0 on S^3

Toda generalized this to his formula for calculating the Hopf inv of a
Toda bracket, which he used e.g. on all the elements I mentioned here.