THE RELATION BETWEEN THE WEISS AND THE GOODWILLIE CALCULI

MARK BEHRENS

These notes are an attempt to conceptually understand the relationship between the Weiss’s orthogonal calculus and Goodwillie’s homotopy calculus.

We give the analogy with functions. We first give the Goodwillie approach. Suppose that we want to study a function f(x) (f(0) = 0) with taylor expansion

f(x) = a_{1}x + a_{2}x^{2} + · · · .

The nth cross effect is defined by

cr_{n}f(x_{1}, …, x_{n}) = f(x_{1} + ⋯ + x_{n})

− ∑_{i} f(x_{1} + ⋯ + x̂_{i} + ⋯ + x_{n})

+ ∑_{i,j} f(x_{1} + ⋯ + x̂_{i} + ⋯ + x̂_{j} + ⋯ + x_{n})

− ⋯

± (f(x_{1}) + ⋯ + f(x_{n}))

We find that when we take the multilinearization of the nth cross effect, (the nth differential) we get

D^{(n)}f(x_{1}, …, x_{n}) = n! a_{n}x_{1} ⋯ x_{n}.

We recover the nth Taylor coefficient by evaluating on the unit.

C_{n}^{G}f := D^{(n)}f(1, …, 1) = n! · a_{n}.

The nth term of the Taylor series is recovered by

D_{n}^{G}f(x) = (C_{n}f · x^{n})/n! = a_{n}x^{n}.

The G in the above notation stands for ”Goodwillie”.

We now give the Weiss approach. Suppose that g(n) is a function on ℕ. Think of it as a sequence that converges rapidly. We could try to understand g(n) by iterated differences. We shall actually do a ‘q-analog’ in some sense. q is a parameter which should be thought of as ”small”.

g_{0}(n) := g(n)
g_{1}(n) := g_{0}(n) − g_{0}(n + 1)
g_{2}(n) := g_{1}(n) − 1/q · g_{1}(n + 1)
g_{3}(n) := g_{2}(n) − 1/q^{2} · g_{2}(n + 1)
⋮

We will define formally the Weiss coeficient to be

C_{k}^{W} g := lim_{n→∞} g_{k}(n)/q^{kn}.

Of course, these limits should have no reason to exist. I’m not sure what I mean by this, but then we should have the nth term of the Weiss tower be given by

(D_{k}^{W} g)(n) = (C_{k}^{W} g · q^{kn})/((1 − q^{k}) ⋯ (1 − q))

and

g(n) = ∑_{n}(D_{n}^{W} g)(n).

It is difficult to imagine a sequence g(n) for which the above analysis could possibly make sense, but here’s one.

g(n) = 1/q^{n} · f(q^{n}x)

Here f(x) is has a taylor expansion as before. Then we have:

g(n) = a_{1}x + q^{n}a_{2}x^{2} + q^{2n}a_{3}x^{3} + ⋯
g_{1}(n) = q^{n}(1 − q)a_{2}x^{2} + q^{2n}(1 − q^{2})a_{3}x^{3} + ⋯
g_{2}(n) = q^{2n}(1 − q^{2})(1 − q)a_{3}x^{3} + q^{3n}(1 − q^{3})(1 − q^{2})a_{4}x^{4} + ⋯
⋮

If q < 1 then the limits that define C_{k}g actually exist, and we have

C_{k}^{W} g = (1 − q^{k}) ⋯ (1 − q)a_{k+1}x^{k+1}

and

(D_{k}^{W} g)(n) = (C_{k}^{W} g · q^{nk})/((1 − q^{k}) ⋯ (1 − q)) = a_{k+1}x^{k+1}q^{kn}.

We have the fundemental relationship

(D_{k}^{W} g)(n) = 1/q^{n} · (D_{k+1}^{G}f)(q^{n}x).

We also view the iterated differences g_{i} above as functions of x and do Goodwillie calculus on these. We have

(D_{n}^{G}g_{m})(x) = a_{n}x^{n} · \frac{(1 − q^{n−1}) ⋯ (1 − q)}{(1 − q^{n−m−1}) ⋯ (1 − q)}.

The thing is that there are analogs of all of these formulas in the actual calculus of Goodwillie and Weiss. I learned these formulas from Greg Arone — they are central to his paper on Mitchell complexes. The Weiss coefficients have an action of the orthogonal group, and the Goodwillie coefficients have an action of the symmetric group. We use the following dictionary.

n! ↔ Σ_{n} (Symmetric group)
q^{n} ↔ Σ^{V} (Suspension)
1/q^{n} ↔ Ω^{V}
(1 − q^{n}) ⋯ (1 − q) ↔ O(V) (Orthogonal group)

If F is a homotopy functor from pointed spaces to pointed spaces for which F (∗) = ∗, then define a functor on vector spaces by

G(V) = Ω^{V} F (Σ^{V} X).

Then, we have the following formulas which relate the terms of the Goodwillie and Weiss towers.

(D_{k}^{W} G)(V) = Ω^{V} (D_{k+1}F)(Σ^{V} X)
(D_{n}^{G}G_{m})(X) = (C_{n}^{G} ∧ X^{∧n}) ∧_{hΣ_{n}} O(n − 1)/O(n − m − 1)

Here, the second formula regards the mth Weiss iterated difference G_{m} as a functor of X, and Σ_{n} is acting on O(n − 1) via the reduced cannonical representation.