Filtrations of global equivariant K-theory

Abstract

Arone and Lesh (J Reine Angew Math 604:73–136, 2007; Fund Math 207(1):29–70, 2010) constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg–MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh’s description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which was computed by Schwede (J Am Math Soc 30(3):673–711, 2017).

Appendix

1.1 Cofibrancy properties of the rank filtration

In this appendix we show that the V-th level of the inclusions $ku^{n-1}\rightarrow ku^n$ is an O(V)-cofibration, guaranteeing that the quotient $ku^n/ku^{n-1}$ has the global homotopy type of the homotopy cofiber. For instance, this was used in the proof of Theorem 3.22. For finite subgroups of O(V) (and hence for the ${\mathcal {F}}in$-global homotopy type of the quotient) this would follow quite directly from the results of [14], but we need the general statement. In this and the next appendix we repeatedly make use of a theorem due to Illman (cf. [9]) that states that every smooth manifold equipped with a smooth action by a compact Lie group allows the structure of an equivariant CW complex.

We recall from [12, Sec. 3] that the evaluation X(A) of a $\Gamma$-space X on a based space A is naturally filtered by skeleta $sk_m(X(A))$. The m-skeleton is obtained from the $(m-1)$-st by forming a certain pushout [12, Thm. 3.10]. Furthermore, given a map $i:X\rightarrow Y$ of $\Gamma$-spaces, one can define relative skeleta $sk_m[i](A)$ by $sk_m(Y(A))\cup _{sk_m(X(A))} X(A)$ and it follows that these are related by a similar pushout square. The colimit over the $sk_m[i](A)$ gives back Y(A) and the map from $X(A)=sk_0[i](A)$ agrees with i. Now let V be a finite dimensional real inner product space. We are interested in the case where A is equal to $S^V$ and i is the inclusion $k^{n-1}({{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}}),-)\hookrightarrow k^n({{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}}),-)$. There the connecting pushout takes the form

(7)

where the wedge is indexed over all m-tuples $(n_1,\ldots ,n_m)$ which add up to n, with all $n_i$ larger than 0. The notation $F((S^V)^{\times m})$ stands for the subspace of $(S^V)^{\times m}$ of tuples which contain two equal entries or a basepoint. It suffices to show that $sk_{m-1}[i](S^V)\rightarrow sk_m[i](S^V)$ is an O(V)-cofibration for all $m\in {\mathbb {N}}$, since the sequential colimit of O(V)-cofibrations is again an O(V)-cofibration. This follows from:

Lemma 7.1

The left hand vertical map in Diagram (7) is an O(V)-cofibration.

Proof

We first argue that $\bigvee _{n_1+\cdots +n_m=n}(L_{\mathbb {C}}(\bigoplus {\mathbb {C}}^{n_i},{{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}}))/{\prod U(n_i)})_+$ is $(\Sigma _m\times O(V))$-cofibrant. This would follow directly from Illman’s theorem if we put $W_k=\bigoplus _{i=0,\ldots ,k}{{\,\mathrm{Sym}\,}}^i(V_{{\mathbb {C}}})$ instead of the full ${{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}})$, since each $L_{\mathbb {C}}(\bigoplus {\mathbb {C}}^{n_i},W_k)$ is a smooth manifold with a smooth action by $U(W_k)\times (N_{U(n)}\prod U(n_i))$. The subspace $L_{\mathbb {C}}(\bigoplus {\mathbb {C}}^{n_i},W_{k-1})$ is exactly the space of $U({{\,\mathrm{Sym}\,}}^k(V_{{\mathbb {C}}}))$-fixed points under this action. Since O(V) fixes ${{\,\mathrm{Sym}\,}}^k(V_{{\mathbb {C}}})$, it normalizes the subgroup $U({{\,\mathrm{Sym}\,}}^k(V_{{\mathbb {C}}}))$. This implies that if we forget any $(U(W_k)\times (N_{U(n)}\prod U(n_i)))$-CW structure to an $(O(V)\times (N_{U(n)}\prod U(n_i)))$-cell structure, the space $L_{\mathbb {C}}(\bigoplus {\mathbb {C}}^{n_i},W_{k-1})$ is necessarily a subcomplex and hence the inclusion a cofibration. By quotienting out the $\prod U(n_i)$-actions and passing to the colimit we see that the wedge is $(\Sigma _m\times O(V))$-cofibrant, as claimed.

Hence it suffices to show that $F((S^V)^{\times m})\rightarrow (S^V)^{\times m}$ is a $(\Sigma _m\times O(V))$-cofibration. Note that $S^V$ is an O(V)-CW complex with the two 0-cells 0 and $\infty$ with trivial O(V)-action and one 1-cell of the form $O(V)/O(V-1)\times D^1$ (where $V-1\subset V$ is a choice of a hyperplane in V). Then the product CW structure on $(S^V)^{\times m}$ has cells of the form

$$\begin{aligned} (\Sigma _m\wr O(V))\ltimes _{\Sigma _i\wr O(V-1)\times \Sigma _j\wr O(V)\times \Sigma _k \wr O(V)} ((D^1)^{\times i}\times \{0\}^{\times j}\times \{\infty \}^{\times k}), \end{aligned}$$

where $i+j+k=m$. These are attached along the inclusions $\partial ((D^1)^{\times i})\rightarrow (D^1)^{\times i}$. They do not quite define a ($\Sigma _n\wr O(V)$)-equivariant CW structure, since $\Sigma _i$ acts non-trivially on the cell $(D^1)^{\times i}$. But after fixing a $\Sigma _i$-equivariant CW structure on $(D^1)^{\times i}$ relative to its boundary for all i one obtains a ($\Sigma _m\wr O(V)$)-equivariant cell structure on $(S^V)^{\times m}$. A further choice of $(\Sigma _m\times O(V))$-CW structure on the $(\Sigma _m\wr O(V))$-orbits then defines a $(\Sigma _m\times O(V))$-cell structure on $(S^V)^{\times m}$. Now, by definition, $F((S^V)^{\times m})$ is the union of two subspaces: the space of tuples containing a basepoint and the space of tuples containing two equal entries. Note that the former is a subcomplex of the $(\Sigma _m\wr O(V))$-cell structure, since the basepoint $\infty$ is a 0-cell, and hence also a subcomplex of the underlying $(\Sigma _m\times O(V))$-cell structure. But the latter is given precisely by those points that have non-trivial $\Sigma _m$-isotropy, hence it is an equivariant subcomplex for any $(\Sigma _m\times O(V))$-cell structure because $\Sigma _m$ is a normal subgroup. Thus, $F((S^V)^{\times m})$ is the union of two $(\Sigma _n\times O(V))$-subcomplexes of $(S^V)^{\times m}$ and hence itself one, and therefore the inclusion is a $(\Sigma _n\times O(V))$-cofibration, as desired. $\square$

1.2 Equivariant CW structures

The content of this appendix is to show that the U(n)-orthogonal spaces ${\overline{{\mathcal {L}}}}_n$ that appeared in Sect. 3.1 give $(U(n)\times G)$-cell complexes when evaluated on any G-representation V (at most countably infinite dimensional). This property was needed in Proposition 3.6 for ${\overline{{\mathcal {L}}}}_n$ to be a global universal space for the family of complete/non-isotypical subgroups of U(n). The same proof also shows that the spaces $\overline{{\mathcal {P}}^R_n}(V)$ arising in the filtrations of algebraic K-theory are $(GL_n(R)\times G)$-cell complexes.

The proof is similar to that of the previous section. This time we consider the (absolute) skeleta filtration for the U(n)-$\Gamma$-space ${\mathcal {L}}(n,-)$, where the relating pushouts take the form

The wedge is taken over the same indexing system as in the previous section. We recall that the closed subspace ${\overline{{\mathcal {L}}}}_n(V)$ of ${\mathcal {L}}(n,S^V)$ was defined to contain those elements that can be represented by a tuple $(W_i,x_i)_{i\in I}$ with all $x_i$ non-equal to the basepoint and satisfying the equations $\sum \dim (W_i)\cdot x_i=0$ and $\sum \dim (W_i)|x_i|^2=1$. Intersection with $sk_m ({\mathcal {L}}(n,S^V))$ gives subspaces $sk_m ({\overline{{\mathcal {L}}}}_n(V))$ whose colimit over m is isomorphic to ${\overline{{\mathcal {L}}}}_n(V)$. Likewise, for fixed $n_1,\ldots ,n_m$ we define closed subspaces $S_{\{n_i\}}((S^V)^{\times m})\subseteq (S^V)^ {\times m}$ as those tuples satisfying $\sum n_i \cdot x_i=0$ and $\sum n_i |x_i|^2=1$. With these definitions an element of $L(\bigoplus {\mathbb {C}}^{n_i},{\mathbb {C}}^ n)/{\prod U(n_i)}\times (S^V)^ {\times m}$ is mapped to $sk_m({\overline{{\mathcal {L}}}}_n(V))$ if and only if it lies in $L(\bigoplus {\mathbb {C}}^{n_i},{\mathbb {C}}^ n)/{\prod U(n_i)}\times S_{\{n_i\}}((S^V)^{\times m})$. So we obtain a new pushout square

(8)

Hence it suffices to show:

Lemma 7.2

The left hand vertical map in Diagram (8) is a $(U(n)\times G)$-cofibration.

Proof

The proof is very similar to that of Lemma 7.1. Again it suffices to see that

$$\begin{aligned} \bigsqcup _{\{n_i\ne 0\}_{0\le i\le m}\sum n_i=n}\left( L\left( \bigoplus {\mathbb {C}}^{n_i},{\mathbb {C}}^ n\right) \Big /{\prod U(n_i)}\right) \end{aligned}$$

is a $(U(n)\times \Sigma _m)$-CW complex and that the map $F(S_{\{n_i\}}((S^ V)^{\times m})))\rightarrow (S^V)^{\times m}$ is a $(\Sigma _m\times G)$-cofibration. The former is easy to see, because each summand is U(n)-isomorphic to $U(n)/\prod U(n_i)$ and these summands are permuted by the $\Sigma _m$-action. For the latter we note that by a transformation of variables each $S_{\{n_i\}}((S^V)^{\times m})$ is homeomorphic to the usual unit sphere $S(V\otimes {\mathbb {R}}^m)$, which—by Illman’s theorem for finite dimensional V and the same trick as in Lemma 7.1 for the infinite case—is a $(\Sigma _m\times G)$-CW complex. Since $F(S_{\{n_i\}}((S^V)^{\times m}))$ no longer contains any basepoints, it is exactly the subspace of elements with non-trivial $\Sigma _m$-isotropy, and hence always a $(\Sigma _m\times G)$-subcomplex. This finishes the proof. $\square$

1.3 Verification of cofiber sequence

In this appendix we give the proof that the map

$$\begin{aligned} \psi _n: \Sigma ^{\infty }_+ (L({\mathbb {C}}^n)\times _{U(n)} {\overline{{\mathcal {L}}}}_n)\rightarrow ku^{n-1} \end{aligned}$$

constructed in Sect. 3.4 makes the following diagram a morphism of triangles in the global homotopy category:

(9)

In order to establish this we turn the upper sequence into a strict quotient sequence by replacing $L({\mathbb {C}}^n)/U(n)$ with $L({\mathbb {C}}^n)\times _{U(n)} C{\overline{{\mathcal {L}}}}_n$, where $C{\overline{{\mathcal {L}}}}_n$ denotes the cone on ${\overline{{\mathcal {L}}}}_n$. We construct a morphism

$$\begin{aligned} {\overline{\psi }}_n:\Sigma ^{\infty }_+ (L({\mathbb {C}}^n)\times _{U(n)} C{\overline{{\mathcal {L}}}}_n)\rightarrow ku^n \end{aligned}$$

with the following three properties:

• (1) The square

  

  commutes.

• (2) The restriction of ${\overline{\psi }}_n$ to the copy of $\Sigma ^{\infty }_+ (L({\mathbb {C}}^n)/U(n))$ at the cone point is equal to $\alpha _n$.

• (3) The induced map

  $$\begin{aligned} \Sigma ^{\infty }(L({\mathbb {C}}^n)_+\wedge ({\overline{{\mathcal {L}}}}_n)^\diamond )\rightarrow ku^n/ku^{n-1}, \end{aligned}$$

  obtained by quotiening out $\Sigma ^{\infty }_+ (L({\mathbb {C}}^n)\times _{U(n)} {\overline{{\mathcal {L}}}}_n)$ and $ku^{n-1}$, is homotopic to the isomorphism constructed in Sect. 3.1.

The first two properties show that ${\overline{\psi }}_n$ induces a homotopy between the two composites in the first square of Diagram 9. The third property implies that there is a homotopy between the two composites in the square

and so we are done. The map ${\overline{\psi }}_n$ is also used in Sect. 3.5.

In order to construct ${\overline{\psi }}_n$ we quickly recall the objects involved: an element of $L({\mathbb {C}}^n)(V)$ is a linear isometry ${\mathbb {C}}^n\hookrightarrow {{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}})$. Points in ${\overline{{\mathcal {L}}}}_n(V)$ are represented by tuples $(W_i,x_i)_{i\in I}$ where the $x_i$ are elements of V and the $W_i$ are pairwise orthogonal subspaces of ${\mathbb {C}}^n$ which add up to all of ${\mathbb {C}}^n$. Furthermore, these tuples have to be reduced and of norm 1 (cf. Sect. 3.1). Finally, elements of $ku^n(V)$ are also represented by tuples $(W_i,x_i)_{i\in I}$, but this time the $W_i$ are orthogonal subspaces of ${{\,\mathrm{Sym}\,}}(V_{{\mathbb {C}}})$ and the only requirement is that the sum of the dimensions is at most n. We recall also that the definition of $\psi _n$ made use of a function $s:[0,\infty ]\rightarrow [0,\infty ]$ which maps the interval $[0,\frac{1}{2n^2}]$ homeomorphically onto $[0,\infty ]$ and is constant on the rest. Finally, given a finite tuple of vectors $x=(x_i)_{i\in I}$ of a real inner product space V we defined a map $p_x:V\rightarrow \langle \{x_i\}_{i\in I} \rangle \subseteq V$ by $p_x(v)=\sum _I \langle v,x_i \rangle \cdot x_i$.

Now let $H:[0,\infty ]\times [0,1]\rightarrow [0,\infty ]$ be a homotopy relative endpoints from the identity to s. Given a real inner product space V with a finite tuple of vectors $x=(x_i)_{i\in I}$ as above, we define a map $H^V_x:S^V\times [0,1]\rightarrow S^V$ via

$$\begin{aligned} H^V_x(v,t)=(H(|p_xv|,t)-|p_xv|)\cdot p_xv + v. \end{aligned}$$

This gives a homotopy from the identity to the map $s^V_x$ used in the definition of $\psi _n$.

Now we can define ${\overline{\psi }}_n$ by the formula

$$\begin{aligned} (\varphi ,(W_i,x_i)_{i\in I},t)\wedge v\mapsto {\left\{ \begin{array}{ll} (\varphi (W_i),\frac{t}{1-t} \cdot x_i+v)_{i\in I} &{} \;\text { if }\; 0\le t\le 1/2\\ (\varphi (W_i),H^V_x(x_i + v,2t-1))_{i\in I} &{} \;\text { if }\; 1/2\le t\le 1 \end{array}\right. } \end{aligned}$$

where x is short for the tuple of the $x_i$. Since $H_x(x_i + v,0)$ is equal to $x_i+v$, these two definitions agree on the intersection and glue to a well-defined map. By definition, setting t equal to 1 gives back $\psi _n$, thus property (1) is satisfied. Furthermore, the elements $(\varphi ,(W_i,x_i)_{i\in I},0)$ are mapped to the tuple $(\varphi (W_i),v)_{i\in I}$, which is equal to $(\varphi ({\mathbb {C}}^n),v)$. Hence it is independent of the $W_i$ and $x_i$ and the induced map

$$\begin{aligned} \Sigma ^{\infty }_+ (L({\mathbb {C}}^n)/U(n))\rightarrow ku^n \end{aligned}$$

equals $\alpha _n$, yielding property (2). It remains to prove property (3), i.e., that the induced map

$$\begin{aligned} {\overline{\psi }}_n':\Sigma ^{\infty }(L({\mathbb {C}}^n)_+\wedge {\overline{{\mathcal {L}}}}_n^\diamond )\rightarrow ku^n/ku^{n-1} \end{aligned}$$

obtained by quotiening out $\Sigma ^{\infty }_+ (L({\mathbb {C}}^n)\times _{U(n)} {\overline{{\mathcal {L}}}}_n)$ and $ku^{n-1}$ is homotopic to the isomorphism from Sect. 3.1. For $t\le 1/2$ the two maps are in fact equal and hence it suffices to construct a homotopy on the part where $t\ge 1/2$, relative to $t=1/2$. This is achieved by the formula

$$\begin{aligned}&(\varphi ,(W_i,x_i)_{i\in I},t,s)\wedge v \mapsto \left[ \left( H^V_x\left( \left( \frac{(1-s)t}{1-(1-s)t}+\frac{s-1}{s+1}+1\right) \cdot x_i\right. \right. \right. \\&\quad \left. \left. \left. +v,s(2t-1)\right) ,\varphi (W_i)\right) _{i\in I}\right] \end{aligned}$$

for $s\in [0,1]$. Continuity is only unclear at points for which $t=1$ and $s=0$, which are mapped to the basepoint. However, by the same estimate as in Sect. 3.4 one sees that the expression $(H^V_x((\frac{(1-s)t}{1-(1-s)t}+\frac{s-1}{s+1}+1)\cdot x_i+v,s(2t-1)),\varphi (W_i))_{i\in I}$ lies in $ku^{n-1}$ already for all t close enough to 1 and s close enough to 0. So the homotopy is actually constant around $s=0$ and $t=1$, hence we are done.