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A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra

Abstract

We show that the homotopy groups of a connective \(\mathbb {E}_k\)-ring spectrum with an \(\mathbb {E}_k\)-cell attached along a class \(\alpha \) in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to \(\alpha \) through degree 2n. Using this, we prove that the \(2n-1\)st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that, after localizing at a prime, \(X(n+1)\) is homotopically unique as the \(\mathbb {E}_1-X(n)\)-algebra with homotopy groups in degree \(2n-1\) killed by an \(\mathbb {E}_1\)-cell. Lastly, we prove analogous theorems for a sequence of \(\mathbb {E}_k\)-ring Thom spectra, for each odd k, which are formally similar to Ravenel’s X(n) spectra and whose colimit is also MU.

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Acknowledgements

Finally, for this author at least, mathematics is a deeply social and communal pursuit. Many people, through e-mails, MathOverflow, and personal conversations, were helpful in understanding much of what follows. Some, but likely not all, of those people are: Bob Bruner, Andrew Salch, Nicolas Ricka, Gabriel Angelini-Knoll, Bogdan Gheorghe, Eric Peterson, Sean Tilson, Omar Antolin-Camarena, Tobias Barthel, Tom Bachmann, Tyler Lawson and Marc Hoyois. The work presented here also benefited immensely from the comments of an anonymous referee.

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Correspondence to Jonathan Beardsley.

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Communicated by Mark Behrens.

Appendix: Some categorical constructions

Appendix: Some categorical constructions

This appendix contains several category theoretic results that are used in the main body of the paper. They will not be surprising to experts, but rigorous proofs of them do not seem to be in the literature, so we include them here. First we describe a version of the bar construction for computing pushouts in quasicategories. Then we investigate the “symmetric powers” filtration on the free \(\mathbb {E}_k\)-algebra on a module over an \(\mathbb {E}_{k+1}\)-ring spectrum.

Lemma 2

Let \(K=N(a\overset{\ell }{\leftarrow }b \overset{r}{\rightarrow }c)\) be the nerve of the span category and \(F:K\rightarrow \mathcal {C}\) be a pushout diagram in a cocomplete quasicategory \(\mathcal {C}\) with \(F(a)=A\), \(F(b)=B\) and \(F(c)=C\). Let \(i:K\rightarrow N(\Delta ^{op})\) be the map of quasicategories induced by the functor \(Span\rightarrow \Delta ^{op}\) taking a and c to [0], b to [1] and the morphisms to \(d_0\) and \(d_1\). Then the quasicategorical left Kan extension of F along i, denoted \(Lan_i(F)\), has the property that \(Lan_i(F)([n])\simeq A\coprod B^{\coprod n}\coprod C\).

Proof

Since \(\mathcal {C}\) is cocomplete, the left Kan extension exists. It follows from [19, Proposition 5.2.4] that Kan extensions are stable under pasting with comma squares. Thus given an object \(x:\Delta ^0\hookrightarrow N(\Delta ^{op})\) we have a commutative diagram:

In the above diagram, the square is a pullback (or comma) square and stability under pullback squares means that on the object \(x\in N(\Delta ^{op})\), the value of \(Lan_i(F)\) can be computed as the colimit over the comma category \(i\downarrow x\) of the composition of the forgetful functor \(U:i\downarrow x\rightarrow K\) followed by F. In other words, on each object, \(Lan_i(F)\) can be computed as the left Kan extension of the composition \({i\downarrow x}\rightarrow K\overset{F}{\rightarrow }\mathcal {C}\) along the map to the final object \(\Delta ^0\). This is essentially the classical “pointwise” description of left Kan extensions. Objects of the category \(i\downarrow x\) are objects of \(N(\Delta ^{op})\) over x in \(N(\Delta ^{op})\) whose domains are i(y) for some \(y\in K\). We will denote them by \(i(y)=[j]\rightarrow x\) for \(y\in \{a,b,c\}\) and \(j\in \{0,1\}\).

To see that this gives the claimed description of \(Lan_i(F)([n])\), we consider the comma category \(i\downarrow [n]\) at the level of objects and 1-morphisms. This suffices because our description of \(Lan_i(F)([n])\) will be as a colimit over a diagram in the nerve of an ordinary category. Note that there is a unique degeneracy map \([0]\rightarrow [n]\) in \(\Delta ^{op}\) corresponding to the unique surjection \([n]\rightarrow [0]\), and that there are n unique degeneracies \([1]\rightarrow [n]\) in \(\Delta ^{op}\) corresponding to the n order-preserving surjections \([n]\rightarrow [1]\) in \(\Delta \). There cannot be any 1-morphisms in \(i\downarrow [n]\) between the n objects \(i(b)=[1]\rightarrow [n]\) because this would require the existence of commutative diagrams implying that some of the n unique surjections in \(\Delta \) be equal. There are however two non-surjective order preserving maps \([n]\rightarrow [1]\) in \(\Delta \) corresponding to the maps that take all the elements of \(\{0,1,\ldots ,n\}\) to either 0 or 1. However, because (the opposites of) the maps \([0]\rightarrow [1]\) are in the image of i, these two non-surjective functions admit 1-morphisms in \(i\downarrow [n]\) to the unique morphisms \(i(a)=[0]\rightarrow [n]\) and \(i(c)=[0]\rightarrow [n]\). This exhausts all the maps \([n]\rightarrow [1]\). Thus, after applying the forgetful functor \(i\downarrow x\rightarrow K\) and then applying F, we have a diagram which is in fact just a coproduct of diagrams:

$$\begin{aligned} F(a\leftarrow b)\coprod \underbrace{F(b)\coprod \cdots \coprod F(b)}_{n~\text {copies}}\coprod F(b\rightarrow c). \end{aligned}$$

The colimit of this diagram in \(\mathcal {C}\) clearly has the desired form. \(\square \)

The above lemma may require some explanation. Recall the classical bar construction \(B_\bullet (M,A,N)\) where M and N are left and right modules over some algebra A, respectively (cf. Sects. 4.1 and 4.2 of [18], especially Example 4.1.2). It is a simplicial object with \(B_n(M,A,N)=M\otimes A^{n}\otimes N\) whose face maps are given by the multiplication on A and the A actions on M and N. In a model category, the bar construction is a way of computing a left derived tensor product \(M\otimes ^L_A N\).

A span in a cocomplete category \(a\overset{r}{\leftarrow }b\overset{\ell }{\rightarrow }c\) induces a left b-module structure on a and right b-module structure on c with respect to the cocartesian monoidal structure. To see this, first notice that every object of such a category is an algebra with respect to this monoidal structure since there is always a “fold” map \(x\coprod x\overset{\phi }{\rightarrow }x\). Then note that the maps r and \(\ell \) of the pushout induce maps \(a\coprod b\overset{1_a\coprod r}{\rightarrow }a\coprod a\overset{\phi }{\rightarrow }a\) and \(b\coprod c\overset{\ell \coprod 1_c}{\rightarrow }c\coprod c\overset{\phi }{\rightarrow }c\). So the colimit of the bar construction in this setting, \(B_\bullet (a,b,c)\), is a model for the homotopy pushout of the diagram \(a\overset{r}{\leftarrow }b\overset{\ell }{\rightarrow }c\).

Thus Lemma 2, using the fact that the colimit of a left Kan extension of F agrees with the colimit of F, can be thought of as showing that a pushout in a quasicategory can be computed by taking the colimit of a suitable bar construction \(B_\bullet (F(a),F(b),F(c))\). In other words, Lemma 2 is just a rephrasing of a standard homotopy theoretical fact in the language of quasicategories.

Now recall from [12, Construction 3.1.3.9] and [12, Construction 3.1.3.13] that the free \(\mathbb {E}_k\)-algebra on an object M of a quasicategory \(\mathcal {C}\) admits a decomposition as a coproduct in \(\mathcal {C}\) of the form \(\coprod _{n\ge 0}Sym_{\mathbb {E}_k}^n(M)\). We will describe the objects \(Sym_{\mathbb {E}_k}^n\) as certain colimits in the proof of Lemma 3. However, if \(\mathcal {O}\) happens to be the \(\infty \)-operad associated to a classical simplicial colored operad \(\mathcal {O}_0\) then \(Sym^n_{\mathcal {O}}(M)\simeq (A[\mathcal {O}_0(n)]\otimes _A M^{\otimes _A n})_{\Sigma _n}\). As such, one recovers the classical symmetric power decomposition of the free \(\mathcal {O}\)-algebra on an object. We will not prove this here but refer the interested reader to [7, Corollary 3.2.7].

Lemma 3

Let A be an \(\mathbb {E}_{k+1}\) ring spectrum and M be an A-module which is d-connective for some \(d\ge 0\). Then \(Sym^n_{\mathbb {E}_k}(M)\) is nd-connective for all \(n>1\).

Proof

The proof is nothing more than a careful interpretation of [12, Construction 3.1.3.9]. We will attempt to transcribe such an interpretation here, where we have replaced \(\mathcal {O}^\otimes \) with \(\mathbb {E}_k^\otimes \), \(\mathcal {C}^\otimes \) with \(LMod_A^\otimes \), C with M, and X and Y with \(\langle 1\rangle \). Recall that the object \(Sym_{\mathbb {E}_k}^n(M)\) is constructed as a colimit of a functor \({\bar{h}}_1:\mathcal {P}(n)\rightarrow LMod_A\), where, in this case, \(\mathcal {P}(n)\) is the full subcategory of \(\mathcal {T}riv^\otimes \times _{\mathbb {E}_k}(\mathbb {E}_k/{\langle 1\rangle })\) spanned by the \((\langle n\rangle ,\alpha :\langle n\rangle \rightarrow \langle 1\rangle )\), with \(\alpha \) an active morphism. Recall that \(\mathcal {T}riv^\otimes \) is the trivial\(\infty \)-operad of [12, Example 2.1.1.20], and active here means that \(\alpha \) is a map of finite pointed sets with \(\alpha ^{-1}(*)=\{*\}\). The 1-simplices of \(\mathcal {P}(n)\) are forced to be precisely the pointed bijections \(\langle n\rangle \rightarrow \langle n\rangle \), so it is a Kan complex. We can project \(\mathcal {P}(n)\) onto \(\mathbb {E}_k^\otimes \) by the composition

$$\begin{aligned} \mathcal {P}(n)\hookrightarrow \mathcal {T}riv^\otimes \times _{\mathbb {E}_k}(\mathbb {E}_k/{\langle 1\rangle })\twoheadrightarrow \mathbb {E}_k/\langle 1 \rangle \twoheadrightarrow \mathbb {E}_k, \end{aligned}$$

and this admits a natural transformation (since it factors through the slice over \(\langle 1\rangle \)) to the constant functor \(h_1:\mathcal {P}(n)\rightarrow \mathbb {E}_k\) valued at \(\langle 1\rangle \). This corresponds to a map \(h:\mathcal {P}(n)\times \Delta ^1\rightarrow \mathbb {E}_k\).

Now because there is (by the definition of \(\infty \)-operad) a coCartesian structure map \(q:LMod_A^\otimes \rightarrow \mathbb {E}_k^\otimes \), we can lift h to a map \({\bar{h}}:\mathcal {P}(n)\times \Delta ^1\rightarrow LMod_A^\otimes \). Then \({\bar{h}}_1\) is defined as the restriction of \({\bar{h}}\) to \(\mathcal {P}(n)\times \{1\}\). The lift is coCartesian over the active map from \(\langle n\rangle \) to \(\langle 1\rangle \), so it must be the monoidal structure map that takes an n-tuple \(\{M_1,\ldots ,M_n\}\in q^{-1}(\langle n\rangle )\simeq LMod_A^{n}\) to \(M_1\wedge _A\cdots \wedge _A M_n\in LMod_A\). In this case, because we are forming the free \(\mathbb {E}_k\)-algebra on M, the image of the unique (up to homotopy) object of \(\mathcal {P}(n)\times \{0\}\) in \(LMod_A^\otimes \) is, by construction, equivalent to \(\{M,M,\ldots ,M\}\in LMod_A^n\). This means that the image of \({\bar{h}}_1\) is equivalent to \(M^{\wedge _A n}\).

Because M is d-connective we have that \(M^{\wedge _A n}\) is nd-connective. By [12, Corollary 4.2.3.5] colimits of A-modules are constructed on underlying spectra. Then since by [12, Proposition 4.4.2.6] any colimit of spectra can be decomposed into coproducts and pushouts, and coproducts and pushouts both preserve connectivity (which can be seen by using, for instance, the Mayer-Vietoris sequence), we know that colimits of A-modules preserve connectivity. Hence we have that the colimit of \({\bar{h}}_1\), which is the definition of \(Sym_{\mathbb {E}_k}^n(M)\), is also nd-connective. \(\square \)

Lemma 4

The functors \(Sym_{\mathbb {E}_k}^n:LMod_A\rightarrow LMod_A\) commute with sifted colimits.

Proof

By the proof of the previous lemma, these functors decompose as first forming the n-fold tensor product of a module M and then taking a colimit of a diagram whose vertices are all equivalent to \(M^{\wedge _A n}\). Taking colimits preserves colimits, so it is only required to show that forming n-fold tensor powers commutes with sifted colimits.

First recall that for any diagram \(F:D\rightarrow LMod(A)\), the tensor product preserves colimits in each variable. In other words, there is an equivalence \(colim_D(F(d))\wedge _A M\simeq colim_D(F(d)\wedge _AM)\). From this fact one can inductively deduce, for any n,

$$\begin{aligned} colim_D(F(d_1))\wedge _A\cdots \wedge _A colim_D(F(d_n))\simeq colim_{D^n}(F(d_1)\wedge _A\cdots \wedge _AF(d_n)). \end{aligned}$$

Now recall that D is sifted if and only if the diagonal map \(\delta :D\rightarrow D\times D\) is cofinal, which implies that the n-fold diagonal \(\delta _n:D\rightarrow D^n\) is also cofinal. That is to say, a colimit over \(D^n\) may be equivalently computed by pulling back to D along \(\delta _n\). Thus it follows that

$$\begin{aligned} colim_{D^n}(F(d_1)\wedge _A\cdots \wedge _AF(d_n))\simeq \mathrm {colim}_D(F(d)^{\wedge n}), \end{aligned}$$

which completes the proof. \(\square \)

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Beardsley, J. A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra. J. Homotopy Relat. Struct. 14, 611–624 (2019). https://doi.org/10.1007/s40062-018-0222-6

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Keywords

  • Quasicategories
  • Thom spectra
  • Complex cobordism
  • n-discs operads
  • Stable homotopy
  • Chromatic homotopy