Naive Z/2-spectrum structure on E smash E?

Question

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it a $\mathbb{Z}/2$-spectrum in the less naive, but still not genuine, sense? (That is, a $\mathbb{Z}/2$-spectrum indexed on the trivial universe.)

I'm thinking of something like the following. I may represent $E$ as an (reduced & continuous) excisive functor from pointed spaces to pointed spaces. Then define

$$G(X) = \mathrm{colim}_{I \times I} \mathrm{Map}(S^{x_1} \wedge S^{x_2}, E(S^{x_1}) \wedge E(S^{x_2}) \wedge X)$$

where $I$ is the category of finite sets and inclusions. Hopefully $G$ is a functor from spaces to $\mathbb{Z}/2$-spaces. If I forget about the $\mathbb{Z}/2$-fixed point set, I can think of it as $E \wedge E$ with its $\mathbb{Z}/2$ action. What spectrum does $G(X)^{\mathbb{Z}/2}$ correspond to? Is there a more familiar name for it? Edit: I seem to be getting $E \vee (E \wedge E)^{h\mathbb{Z}/2}$, but without much confidence.

[Leftover part of the question: If so, by my question here I can think of the resulting object as a functor from the opposite of the orbit category of $\mathbb{Z}/2$ to spectra. Unpacking this amounts to giving some spectrum $F$ together with a map $F \to (E \wedge E)^{h\mathbb{Z}/2}$. What is $F$?]

Answer 1

Given a spectrum $E$ there is a "standard" lift of $E \wedge E$ to a $\mathbb{Z}/2$-spectrum using the basic technique you describe. One way to describe it as follows.

You can construct the category of genuine $\mathbb{Z}/2$-spectra (indexed on the full universe) via collections of $\mathbb{Z}/2$-spaces $X\_n$ with equivariant structure maps $\sigma: S^V \wedge X\_n \to X\_{n+1}$, where $S^V = S^1 \wedge S^1$ is the 1-point compactification of the regular representation $\mathbb{R} \times \mathbb{R}$ with the "flip" action. Under this description, if $E$ is a spectrum made up of spaces $E\_n$ and structure maps $S^1 \wedge E\_n \to E\_{n+1}$, then you can construct $E \wedge E$ as a genuine $\mathbb{Z}/2$-spectrum with spaces $E\_n \wedge E\_n$ and structure maps $S^1 \wedge S^1 \wedge E\_n \wedge E\_n$ that simply twist and apply the structure map on each factor. (This, e.g., is one way to pass forward the equivariant structure on $TC$).

This fully genuine spectrum has an underlying spectrum indexed on the trivial subgroup. The $\mathbb{Z}/2$-fixed point object is the homotopy pullback of a diagram

$$ (E \wedge E)^{h\mathbb{Z}/2} \to (E \wedge E)^{t\mathbb{Z}/2} \leftarrow E $$

where $Z^{t\mathbb{Z}/2}$ is the so-called "Tate spectrum" of $Z$, which is to Tate cohomology as the homotopy fixed point spectrum is to group cohomology.

If $E = \Sigma^\infty W$ for a space $W$ then the map from $E$ to the Tate spectrum lifts (via the diagonal) to a map to the homotopy fixed point spectrum, and so the homotopy pullback will actually be homotopy equivalent to $E \vee (E \wedge E)_{h\mathbb{Z}/2}$. (This homotopy orbit is the fiber of the map from homotopy fixed points to Tate fixed points.)

For finite spectra the map from $E$ to its Tate spectrum is 2-adic completion; this is one way to state the content of the Segal conjecture that Carlsson proved (at least at the prime 2). Sverre Lunoe-Nielsen extended this result to a number of other spectra like the Brown-Peterson spectra. In these cases the fixed-point object is equivalent after 2-adic completion to the homotopy fixed point object.

All the above plays out the same way for a cyclic group of prime order.

Answer 2

Here's a very simple way to obtain $(E \wedge E)^{\Bbb Z_2}$ without having to resort to representations (at least if $E$ is connective). Consider the functor from spectra to spectra given by $$E \mapsto \Sigma^\infty \Omega^\infty E .$$ That is, the suspension spectrum of the zeroth space of $E$.

Let $$E \mapsto P_2(E)$$ be its quadratic approximation in the sense of Goodwillie's calculus of homotopy functors (that is, the second stage of the Goodwillie tower).

Then $P_2(E)$ has the homotopy type of $(E \wedge E)^{\Bbb Z_2}$ whenever $E$ is connective.

Incidentally, one can also see that this comes equipped with a fiber sequence $$ D_2(E) \to P_2(E) \to E $$ which amounts to the tom Dieck splitting in the case when $E$ is a suspension spectrum. Here, $D_2(E) = (E\wedge E)_{h\Bbb Z_2}$ is the quadratic construction. In general, this sequence needn't split.