Band width and the Rosenberg index

The author would like to thank Shinichiroh Matsuo for a helpful comment. This work was supported by RIKEN iTHEMS and JSPS KAKENHI Grant Numbers 19K14544, JPMJCR19T2, 17H06461.

The first step for the proof is an observation on the systole, the minimum of the length of homotopically non-trivial loop, of a subspace of a uniformly $1$-connected manifold.

Let $M$ be a $d$-dimensional complete Riemannian spin manifold with infinite $\mathcal{KO}$-width. For each $n\in\mathbb{N}$, pick a compact Riemannian band $V_{n}$ in the class $\mathcal{KO}$ whose width is greater than $n$, and an isometric immersion $f_{n}\colon V_{n}\to M$. Let $\pi_{n}:=\pi_{1}(V_{n})$ and let $\Gamma_{n}\subset\Gamma$ denote the image of $\pi_{n}$ under the inclusion $V_{n}\to M$. Then the map $f_{n}$ lifts to a codimension $0$ immersion $\tilde{f}_{n}\colon\widetilde{V}_{n}\to\widetilde{M}$, where $\widetilde{V}_{n}$ is the universal covering of $V_{n}$. In short, we write $\mathbf{V}$ and $\widetilde{\mathbf{V}}$ for the disjoint union $\bigsqcup_{n\in\mathbb{N}}V_{n}$ and $\bigsqcup_{n\in\mathbb{N}}\widetilde{V}_{n}$ respectively. For $R>0$, let $\widetilde{V}_{n,R}$ denote the open subset of $\widetilde{V}_{n}$ consisting of points $x\in\widetilde{V}_{n}$ such that $d(x,\partial\widetilde{V})\geq R$.

Let $\widetilde{U}_{n}$ and $\widetilde{U}_{n,R}$ denote the interior of $\tilde{f}(\widetilde{V}_{n})$ and $\tilde{f}(\widetilde{V}_{n,R})$ respectively, which are non-empty open subsets of $\widetilde{M}$. For an inclusion $Y\subset X$ of length spaces, we call the infimum of the length of closed loops in $Y$ representing non-trivial homotopy class in $X$ the relative systole of $Y\subset X$ and write $\mathrm{sys}(Y\subset X)$. This corresponds to the systole of $Y$ relative to $\pi_{1}(Y)\to\pi_{1}(X)$ in the standard terminology of systolic geometry (cf. [katzSystolicGeometryTopology2007]*Definition 8.2.1).

The following lemma is standard in coarse geometry, rather known as the uniform contractibility of the universal covering of an aspherical manifold (see e.g. [roeLecturesCoarseGeometry2003]*Example 5.26).

Next, we construct a lift of an operator on $\widetilde{M}$ to the non-compact Riemannian bands $\widetilde{V}_{n}$, which forms a $\ast$-homomorphism ‘modulo boundary’. This technique is inherited from [kubotaCodimensionTransferHigher2021].

Let $T\in\mathbb{B}(L^{2}(\widetilde{M}))$ which is locally Hilbert–Schmidt, i.e., $Tf$, $fT$ is in the Hilbert–Schmidt class for any $f\in C_{c}(\widetilde{M})$. Such $T$ is represented by a kernel function $T\colon\widetilde{M}\times\widetilde{M}\to\mathbb{C}$ as

The support of $T$ is defined as the support of its kernel function in $\widetilde{M}\times\widetilde{M}$, and $\mathrm{Prop}(T):=\sup\{d(x,y)\mid(x,y)\in\operatorname{\mathrm{supp}}T\}$ is called the propagation of $T$, where $d$ denotes the metric on $\widetilde{M}$. We define $\mathbb{C}_{\mathrm{HS}}[\widetilde{M}]^{\Gamma}$ as the Real $\ast$-algebra of $\Gamma$-invariant locally Hilbert–Schmidt operators with finite propagation. Let $C^{*}(\widetilde{M})^{\Gamma}$ denote the completion of $\mathbb{C}_{\mathrm{HS}}[\widetilde{M}]^{\Gamma}$ with respect to the maximal norm satisfying the C*-condition $\|T^{*}T\|=\|T\|^{2}$ (which is well-defined by [gongGeometrizationStrongNovikov2008]*3.5). In the same way, we also define $\mathbb{C}_{\mathrm{HS}}[\widetilde{V}_{n}]^{\pi_{n}}\subset\mathbb{B}(L^{2}(\widetilde{V}_{n}))$ and its completion $C^{*}(\widetilde{V}_{n})^{\pi_{n}}$.

We define the Real $\ast$-algebra $\mathbb{C}_{\mathrm{HS}}[\widetilde{\mathbf{V}}]^{\boldsymbol{\pi}}$ consisting of sequences of locally compact operators $T_{n}\in\mathbb{B}(L^{2}(\widetilde{V}_{n}))$ with a uniform bound of propagation, i.e.,

and let $C^{*}(\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ denote its closure in the direct product C*-algebra $\prod_{n\in\mathbb{N}}C^{*}(\widetilde{V}_{n})^{\pi_{n}}$. We also define the Real C*-ideal $C^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}\triangleleft C^{*}(\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ as the closure of

where $d(\operatorname{\mathrm{supp}}T_{n},\partial\widetilde{V}_{n}\times\partial\widetilde{V}_{n})$ stands for the Hausdorff distance.

Let $D^{*}(\widetilde{M})^{\Gamma}$ denote the closure of the set of bounded operators on $L^{2}(\widetilde{M})$ which is of finite propagation and is pseudo-local, i.e., $[T,f]\in\mathbb{K}$ for any $f\in C_{c}(\widetilde{M})$, with respect to the norm

This is a Real C*-algebra [oyono-oyonoTheoryMaximalRoe2009]*Lemma 2.16, including $C^{*}(\widetilde{M})^{\Gamma}$ as a Real C*-ideal. We write the quotient as $Q^{*}(\widetilde{M})^{\Gamma}:=D^{*}(\widetilde{M})^{\Gamma}/C^{*}(\widetilde{M})^{\Gamma}$. A standard fact in coarse index theory is that the K-group of $Q^{*}(\widetilde{M})^{\Gamma}$ is isomorphic to the equivariant K-homology $\operatorname{\mathrm{KO}}^{\Gamma}_{*}(\widetilde{M})$. In the same way, we also define the Real C*-algebras $D^{*}(\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$, $Q^{*}(\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$, $D^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$, and $Q^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$. We only note that $D^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ is the closure of the set of pseudo-local, finite propagation operators which is supported near $\partial V$ and $Tf,fT\in\mathbb{K}$ for any $f\in C_{c}(\mathbf{V}\setminus\partial\mathbf{V})$.

In the last step, we relate the equivariant coarse index of $\widetilde{M}$ with that of $\partial\widetilde{V}_{n}$ through the lifting homomorphism $s$ constructed above.

We define the ideal $C^{*}_{0}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ of $C^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ consisting of operators $(T_{n})_{n}$ such that $\|T_{n}\|\to 0$ as $n\to\infty$, and set

We also define the Real C*-algebras $D^{*}_{\flat}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ and $Q^{*}_{\flat}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}$ in the same way.

We write $\phi_{\pm}$ for the first and the second component of $\phi$ respectively.

Let

denote the K-theory boundary map associated to the exact sequence

We also use the same letter $\partial$ for the $\operatorname{\mathrm{KO}}$-theory boundary map of the same kind defined for $D^{*}$ and $Q^{*}$ coarse C*-algebras.

Let $\not{D}_{\widetilde{M}}$, $\not{D}_{\widetilde{V}_{n}}$, $\not{D}_{\partial_{\pm}\widetilde{V}_{n}}$ denote the $\text{\it C$\ell$}_{d,0}$-linear Dirac operator on $\widetilde{M}$ and $\widetilde{V}_{n}$, and the $\text{\it C$\ell$}_{d-1,0}$-linear Dirac operator on $\partial_{\pm}\widetilde{V}_{n}$ respectively (for the definition of $\text{\it C$\ell$}_{d,0}$-linear Dirac operator on a $d$-dimensional spin manifold, see e.g. [lawsonjr.SpinGeometry1989]*Chapter II, §7). We also consider the Clifford-linear Dirac operators $\not{D}_{\widetilde{\mathbf{V}}}$ and $\not{D}_{\partial_{\pm}\widetilde{\mathbf{V}}}$, each of which is the same thing as the family $(\not{D}_{\widetilde{V}_{n}})_{n\in\mathbb{N}}$ and $(\not{D}_{\partial_{\pm}\widetilde{V}_{n}})_{n\in\mathbb{N}}$ respectively. Let $\chi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function such that $\chi(t)\to\pm 1$ as $t\to\pm\infty$ and the support of $\hat{\chi}$ is in $(-1,1)$. Then $\chi(\not{D}_{\widetilde{M}})$ is an odd self-adjoint operator in $D^{*}(\widetilde{M})^{\Gamma}$ by [roePartitioningNoncompactManifolds1988]*Proposition 2.3, whose image in $Q^{*}(\widetilde{M})^{\Gamma}$ is unitary. Hence it determines an element of $\operatorname{\mathrm{KO}}_{d+1}(Q^{*}(\widetilde{M})^{\Gamma})$ (cf. [kubotaCodimensionTransferHigher2021]*Remark A.1), which is denoted by $[\not{D}_{\widetilde{M}}]$ in short. Similarly, we define $[\not{D}_{\partial_{\pm}\widetilde{V}_{n}}]\in\operatorname{\mathrm{KO}}_{d}(Q^{*}(\partial_{\pm}\widetilde{V}_{n})^{\pi_{n}})$, and $[\not{D}_{\partial\widetilde{\mathbf{V}}}]\in\operatorname{\mathrm{KO}}_{d}(Q^{*}(\partial\widetilde{\mathbf{V}})^{\boldsymbol{\pi}})$. We also define the relative KO-class of the Dirac operators on manifolds with boundary $[\not{D}_{\widetilde{V}_{n}}]\in\operatorname{\mathrm{KO}}_{d+1}(Q^{*}(\widetilde{V}_{n})^{\pi_{n}}/Q^{*}(\partial\widetilde{V}_{n}\subset\widetilde{V}_{n})^{\pi_{n}})$ and $[\not{D}_{\widetilde{\mathbf{V}}}]\in\operatorname{\mathrm{KO}}_{d+1}(Q^{*}(\widetilde{\mathbf{V}})^{\boldsymbol{\pi}}/Q^{*}(\partial\widetilde{\mathbf{V}}\subset\widetilde{\mathbf{V}})^{\boldsymbol{\pi}})$.

The K-theory boundary map of the extensions $0\to C^{*}(X)^{G}\to D^{*}(X)^{G}\to Q^{*}(X)^{G}\to 0$ are denoted by $\operatorname{\mathrm{Ind}}_{G}$, where $(X,G)$ is $(\widetilde{M},\Gamma)$, $(V_{n},\pi_{n})$, $(\widetilde{\mathbf{V}},\boldsymbol{\pi})$, and so on. Note that $C^{*}(M)^{\Gamma}$ is Morita equivalent to the maximal group C*-algebra $C^{*}\Gamma$ and the Rosenberg index $\alpha_{\Gamma}(M)$ is the same thing with $\operatorname{\mathrm{Ind}}_{\Gamma}([\not{D}_{\widetilde{M}}])$.

Hereafter, for a sequence of abelian groups $A_{n}$ and an element $(a_{n})_{n}\in\prod A_{n}$, we write $(a_{n})_{n}^{\flat}$ for its image by the quotient map $\prod A_{n}\to\prod A_{n}/\bigoplus A_{n}$.