X_1
| |
v v
X_2 ---> X_{12}
is a homotopy pushout and the map X-->X_i is k_i-connected, then the
canonical map
X --> holim ( X_1 --> X_{12} <-- X_2 )
is k-connected where k = k_1 + k_2 - 1 .
Notice how have I changed the statement:
(1) I made it a square rather than a triad and I wrote k_1 and k_2 for p-1
and q-1.
(2) I no longer assume that the spaces are simply connected or that the
k_i are positive.
(3) I assume connectivity of the maps X-->X_1 and X-->X_2 rather than the
maps X_2-->X_{12} and X_1-->X_{12}. (This would have made no difference if
it were not for (2), but as it is it is a necessary strengthening of the
hypothesis.)
(4) I do not explicitly mention triad homotopy groups. They appear here
implicitly as follows: the \pi_j of the triad is \pi_{j-1} of the homotopy
fiber (or rather of some homotopy fiber) of the map
X --> holim ( X_1 --> X_{12} <-- X_2 )
The point about (4) is that I don't know a definition of triad homotopy
groups under which their vanishing will tell us that the map
X --> holim ( X_1 --> X_{12} <-- X_2 )
induces a surjection of \pi_0 . So now we come to the soapbox talk.
SOAPBOX
If you have to deal with spaces that might not be path-connected, you have
to be careful how you define k-connectedness of a pair, or of a map. When
I call a space k-connected, I mean that, for every j with -1\leq j\leq k,
every map from S^j to X extends to D^{j+1}. So for example the empty space
is not k-connected if k \geq -1. When I say that a pair, or more generally
a map X-->Y, is k-connected, I mean that for every point in Y the homotopy
fiber (a.k.a. mapping fiber, a.k.a. holim ( * <-- X --> Y ) ) is a
(k-1)-connected space. Here is where you have to be careful: If the spaces
are path-connected (= 0-connected), then k-connected map means vanishing
of \pi_j(pair) for 1\leq j\leq k. But in general k-connected map means
vanishing of those (for every base point in X) plus surjectivity of \pi_0.
Yes, you can get around this by making an (in my view artificial)
definition of relative \pi_0. You can say relative \pi_0(X-->Y) means the
obvious quotient of \pi_0(Y). But this doesn't do the job if X is empty;
you should instead say \pi_0(X-->Y)=colim(*<--\pi_0(X)-->\pi_0(Y)). And
you'll still probably get it wrong when you say "k-connected map means
vanishing of relative \pi_j for 0\leq j\leq k", because for j>0 you have
to say "for all basepoints in X" while for j=0 you had better not say that
(again, what if X is empty?)
All right, I know I can get a little nutty about the empty set. But there
are reasons. I don't believe that the 0-connectedness of the map
X --> holim ( X_1 --> X_{12} <-- X_2 )
for a square can reasonably be expressed as the vanishing of some kind of
"triad \pi_{-1}". And, while many homotopy theorists may not have much
occasion to think about spaces that are not path-connected, that has not
been my fate; in some of my favorite applications of these definitions and
results, I want to prove that a space X that is the "first" space in a
cube is nonempty by using connectivity information about the cube and
about the other spaces.
ENDSOAPBOX
Anyway, the Blakers-Massey proof of triad connectivity used homology, but
there is a more direct argument using general position that yields the
result as stated above. I can't remember where I saw it first, but as Bill
Richter mentioned some such proof is given in Brayton Gray's book.
Terminology: I like to call a square "k-cartesian" if that map
X --> holim ( X_1 --> X_{12} <-- X_2 )
is k-connected. I also like to call it "k-cocartesian" if the analogous
map
hocolim ( X_1 <-- X --> X_2 ) ---> X _{12}
is k-connected. So infty-cocartesian means homotopy pushout and
infty-cartesian means homotopy pullback.
[Both of these notions -- k-cartesian square and k-cocartesian square --
seem to have an equal claim to the name "k-connected square". But even
when k=infty they are two different notions (that's the whole point!), so
we just don't use that name.]
Note that the theorem discussed above says that infty-cocartesian implies
(k_1 + k_2 -1)-cartesian if the maps in the square are k_i-connected.
There is an easier "dual" result to the effect that infty-cartesian
implies (k_1 + k_2 +1)-cocartesian if the maps in the square are
k_i-connected. (The proof boils down to the fact that the join of a
p-connected and a q-connected space is (p+q+2)-connected. Note that this
is valid even when p and/or q is not positive.)
Now, briefly, about the generalizations from squares (triads) to n-cubes
((n+1)-ads).
Let's call a CW (n+1)-ad (A;A_1,...,A_n) "complete" if A=A_i\cup A_j for
all i and j. These (n+1)-ads correspond to n-cubes which are "strongly
infty-cocartesian" in the sense that every square face of the cube is
infty-cocartesian. Think of these as the ones you get by taking n
cofibrations X-->X_i sharing a common domain and making a cube by pushout
from there.
Let's call an n-cube k-cartesian if the canonical map from the first space
to the holim of the others is k-connected, and call it k-cocartesian if
the canonical map to the last space from the hocolim of the others is
k-connected.
(By the way "strongly infty-cocartesian" implies that every >2 dimensional
face of the cube, in particular the whole n-cube if n>1, is
infty-cocartesian.)
Here is the result that I am accustomed to calling the Higher
Blakers-Massey Theorem, and that Bill Richter called the "n-ad
Connectivity Theorem" in his recent post:
HBM: If a strongly infty-cocartesian n-cube is such that for each i from 1
to n the map X-->X_i from the first space to its ith neighbor is
k_i-connected, then the cube is k-cartesian where
k = k_1 + ... + k_n - n + 1 .
My personal history with this result is as follows.
When I first became interested in multirelative connectivity questions (in
the late 70's, as a student, thinking about spaces of smooth embeddings)
and was looking for tool like HBM above, somebody steered me to the
Barratt-Whitehead paper "The first non-vanishing group of an (n+1)-ad".
B-W stated both connectivity and computation of critical group. It stayed
away from \pi_1 (and \pi_0) difficulties in the same way as the original
Blakers-Massey result. I suppose it used homology, but I have not looked
at the paper in a long time.
I had trouble reading that paper. (I have since heard others say that they
couldn't figure it out either.) So I reproved the connectivity result
myself, in the form stated above as HBM.
My method was the same general position method mentioned above, plus
induction on n, plus some complicated bookkeeping.
Later, when I finally published my "Calculus" papers, I included that
proof (in section 2 of Calculus 2). In the meantime I had also learned
about yet another proof, due to Ellis and Steiner, using the cat^n stuff,
so of course I referenced that.
There is also the dual result:
HBM*: If a strongly infty-cartesian n-cube is such that for each i the map
to the last space from its ith neighbor is k_i-connected, then the cube is
k-cocartesian where k = k_1 + ... + k_n + n - 1 .
This is also proved in Calculus 2, but the proof does not require
induction on n (or complicated bookkeeping).
What I meant by Higher Blakers-Massey theorems in my post about
cosimplicial spaces was a family of results that all appear in section 2
of my Calculus 2 paper:
- The theorem called HBM above.
- The dual result called HBM* above.
- A more flexible and elaborate variant of HBM, that applies to cubes that
are not necessarily strongly infty-cocartesian.
- A similar variant of HBM*.
Let me say a few words about the latter two.
It follows easily from the triad connectivity theorem (i.e. the case n=2
of HBM) that if you have a square in which X-->X_1 is k_1-connected,
X-->X_2 is k_2-connected, and the square is k_{12}-cocartesian, then the
square is also k-cartesian where
k=min(k_1 + k_2 -1, k_{12} - 1). The basic idea is (wlog the square is
cofibrant and then) look at the two squares
X ---> X_1 = X_1
| | |
v v v
X_2 --> colim(X_1<-X->X_2) --> X
The left one is (k_1 + k_2 -1)-cartesian by triad connectivity; the right
one is (k_{12] - 1)-cartesian by the other hypothesis.
That's the more flexible version of HBM when n=2. For n=3 the statement is
that if you have a 3-cube and you know how cocartesian it is, as well as
how cocartesian the three square "faces"
X ---> X_i
| |
v v
X_j ---> X_{ij}
are, and also how connected the three maps X-->X_i are, then you can write
down a lower bound for how cartesian the whole 3-cube is.
And so on.
This is really an easy corollary of HBM. The complicated "diagram chase"
that Richter referred to was just my rather opaque way of writing down the
deduction of the corollary from the HBM theorem.
But there's this other wrinkle: when I proved HBM I had to simultaneously
prove the corollary, because I needed the corollary for n-1 to prove the
theorem for n.
The point is this: The general-position argument leads from an n-cube to
an (n-1)-cube. For example, given a square
X ---> X_1
| |
v v
X_2 ---> X_{12} = colim(X_1<-X->X_2)
where X->X_i are high-dimensional CW pairs, it tells you how connected the
induced map
hf(X-->X_1) ---> hf(X_2-->X_{12})
of homotopy fibers is, which is really what you want to know when you are
asking how cartesian the square is. When I generalized this to a strongly
infty-cocartesian n-cube, I was led to an (n-1)-cube of homotopy fibers
that was not strongly infty-cocartesian, but for which I could read off
the sort of information required by the corollary.
By the way, my interest in the corollary (the "more flexible and elaborate
version") of HBM goes beyond the fact that I found myself needing it to
prove HBM; it's really an indispensable tool for dealing with examples.
I don't have anything to add to the discussion of calculation of critical
group in the general (nonabelian) case.
In the abelian case, I mean in the case of a strongly infty-cocartesian
n-cube (<--> complete (n+1)-ad) whose numbers k_i are all at least 2, you
can easily use HBM for n+1 to reduce the computation of the critical group
to a special case where the answer is given by, for example, a small part
of the Hilton-Milnor Theorem.
This is all related to my Taylor tower idea, if you like, and from one
point of view the "critical group" information in the abelian case is the
tip of an iceberg called the nth derivative of the identity.
Tom Goodwillie
________________________________________________________________
Subject: Re: n-ad connectivity and critical groups
From: Bill Richter