---
abstract: 'We study the de Rham complex on a smooth manifold with a periodic end modeled on an infinite cyclic cover $\tilde X \to X$. The completion of this complex in exponentially weighted $L^2$-norms is Fredholm for all but finitely many exceptional weights determined by the eigenvalues of the covering translation map $H_*(\tilde X) \to H_*(\tilde X)$. We calculate the index of this weighted de Rham complex for all weights away from the exceptional ones.' 
address:
- 'Department of MathematicsMassachusetts Institute of Technology Cambridge, MA 02139'
- 'Department of Mathematics, MS 050Brandeis University Waltham, MA 02454'
- 'Department of MathematicsUniversity of Miami, PO Box 249085 Coral Gables, FL 33124'
author:
- Tomasz Mrowka
- Daniel Ruberman
- Nikolai Saveliev
title: Index theory of the de Rham complex on manifolds with periodic ends
---

Introduction
============

Let $M$ a smooth closed orientable manifold of dimension $n$. The de Rham complex of complex valued differential forms on $M$, $$\begin{CD}
0 \to \Omega^0 (M) @>d_0 >> \Omega^1 (M) @>d_1 >> \Omega^2 (M) \to\; \cdots\; \to \Omega^n (M) \to 0,
\end{CD}$$ is known to be Fredholm in a suitable $L^2$ completion.  This means as usual that the images of $d_k$ are closed and the vector spaces $\ker d_k/{\operatorname{im}}d_{k-1}$ are finite dimensional.  The alternating sum of the dimensions of these spaces is called the index of the de Rham complex.  Since $\ker d_k/{\operatorname{im}}d_{k-1}$ is isomorphic to the singular cohomology $H^k (M;{\mathbb C})$ by the de Rham theorem, the above index equals $\chi (M)$, the Euler characteristic of $M$.

This paper extends these classical results to certain non-compact manifolds, those with a periodic end modeled on an infinite cyclic cover $\tilde X$ of a closed manifold $X$. It builds on the earlier work of Miller [@miller] and Taubes [@taubes:periodic] and can be viewed as a continuation of our research in [@mrs1] and [@mrs2] on the index theory of elliptic operators on such manifolds. 

Let $M$ be a Riemannian manifold with periodic end modeled on $\tilde X$. Its de Rham complex can be completed in the $L^2$ norm using (on the end) a Riemannian measure $dx$ lifted from that on $X$. This completion is, however, not Fredholm; see Remark \[R:fred\].  To rectify this problem, we will use $L^2_{\delta}$ norms, which are the $L^2$ norms on $M$ with respect to the measure $e^{\delta f(x)}\,dx$ over the end.  Here, $\delta$ is a real number and $f: \tilde X \to {\mathbb R}$ is a smooth function such that $f(\tau(x)) = f(x) + 1$ with respect to the covering translation $\tau: \tilde X \to \tilde X$. The $L^2_{\delta}$ completion of the de Rham complex on $M$ will be denoted by $\Omega^*_{\delta} (M)$.

\[T:one\] Let $M$ be a smooth Riemannian manifold with a periodic end modeled on $\tilde X$, and suppose that $H_* (M;{\mathbb C})$ is finite dimensional.  Then $\Omega^*_{\delta} (M)$ is Fredholm for all but finitely many $\delta$ of the form $\delta = \ln|\lambda|$, where $\lambda$ is a root of the characteristic polynomial of $\tau_*: H_* (\tilde X;{\mathbb C}) \to H_* (\tilde X;{\mathbb C})$.

Conditions on $X$ that guarantee that $H_* (M;{\mathbb C})$ is finite dimensional can be found in Section \[S:fred\], together with a proof of Theorem \[T:one\]. 

Given a manifold $M$ as in the above theorem, the complex $\Omega^*_{\delta} (M)$ has a well defined index ${\operatorname{ind}}_{\delta} (M)$. Miller [@miller] showed that ${\operatorname{ind}}_{\delta} (M)$ is an even or odd function of $\delta$ according to whether $\dim M = n$ is even or odd, and that ${\operatorname{ind}}_{\delta} (M) = (-1)^n\,\chi (M)$ for sufficiently large $\delta > 0$. We add to this knowledge the following result. 

\[T:two\] Let $M$ be as in Theorem \[T:one\].  Then ${\operatorname{ind}}_{\delta} (M)$ is a piecewise constant function of $\delta$ whose only jumps occur at $\delta = \ln |\lambda|$, where $\lambda$ a root of the characteristic polynomial $A_k(t)$ of $\tau_*: H_k (\tilde X;{\mathbb C}) \to H_k (\tilde X;{\mathbb C})$ for some $k = 0,\ldots,n-1$. Every such $\lambda$ contributes $(-1)^{k+1}$ times its multiplicity as the root of $A_k(t)$ to the jump. 

Together with the results of [@miller] this completes the calculation of the function ${\operatorname{ind}}_{\delta} (M)$. Theorem \[T:two\] is proved in Section \[S:index\].  The last section of the paper contains discussion as well as calculations of ${\operatorname{ind}}_{\delta} (M)$ for two important classes of examples.  The first class consists of manifolds with infinite cylindrical ends studied earlier by Atiyah, Patodi and Singer [@aps:I], and the second of manifolds arising in the study of knotted $S^2 \subset S^4$.

Finally, we will remark that the de Rham complex is a special case of the more general concept of an elliptic complex.  The index theory for elliptic complexes on closed manifolds was developed by Atiyah and Singer whose famous index theorem [@atiyah-singer:I] expresses the index of such a complex in purely topological terms.  More generally, Atiyah, Patodi and Singer [@aps:I] computed the index for certain elliptic operators (that is, elliptic complexes of length two) on manifolds with cylindrical ends.  In our paper [@mrs2] we extended their result to general manifolds with periodic ends; our index formula involves a new end-periodic $\eta$-invariant, generalizing the $\eta$-invariant of Atiyah, Patodi and Singer from the cylindrical setting.  It would be interesting to compare the index formula of Theorem \[T:two\] with that of [@mrs2] for the operator $d + d^*$ obtained by wrapping up the de Rham complex. 

**Acknowledgments:** We thank Dan Burghelea and Andrei Pajitnov for enlightening discussions on this material. 

The Fredholm property {#S:fred}
=====================

In this section, we will prove Theorem \[T:one\] by reducing it to a statement about twisted cohomology of $X$.

The Fourier–Laplace transform
-----------------------------

The de Rham complex of $M$ is an elliptic complex on the end-periodic manifold $M$ hence we can use the general theory of such complexes due to Taubes [@taubes:periodic].  According to that theory, it is sufficient to check the Fredholm property of the de Rham complex of $\tilde X$ completed in the $L^2$ norm on $\tilde X$ with respect to the measure $e^{\delta f(x)}\,dx$. The latter complex can be studied using the Fourier–Laplace transform, which is defined by the formula $$\hat\omega_z \;=\;\sum\;z^k\cdot (\tau^*)^k\;\omega,\quad z \in {\mathbb C}^*,$$ on compactly supported forms $\omega$ on $\tilde X$, and is extended by continuity to $L^2$ forms.  The summation in the above formula extends to all integer $k$, which makes the form $\hat\omega_z$ invariant with respect to $\tau^*$. The form $\hat\omega_z$ then defines a form on $X$ which is denoted by the same symbol.  An application of the Fourier–Laplace transform to the de Rham complex on $\tilde X$ results in a family of twisted de Rham complexes $$\label{E:twist}
\begin{CD}
\cdots @>>> \Omega^k (X) @> \quad d - \ln z\,df\quad >> \Omega^{k+1} (X) @>>> \ldots
\end{CD}$$ parameterized by $z \in {\mathbb C}^*$. The following result is proved in Taubes [@taubes:periodic Lemma 4.3]. 

\[P:taubes\] Let $M$ be a smooth Riemannian manifold with a periodic end modeled on $\tilde X$. For any given $\delta$, the complex $\Omega^*_{\delta} (M)$ is Fredholm if and only if the complexes are exact for all $z$ such that $|z| = e^{\delta}$.

The cohomology of complex is of course the twisted de Rham cohomology $H^*_z (X;{\mathbb C})$ with coefficients in the complex line bundle with flat connection $-\ln z\,df$.

\[P:fred\] Let $M$ be a smooth Riemannian manifold with a periodic end modeled on $\tilde X$. Then the following three conditions are equivalent:

1.   $\Omega^*_{\delta} (M)$ is Fredholm for all $\delta \in {\mathbb R}$ away from a discrete set;

2.   $H^*_z (X;{\mathbb C})$ vanishes for all $z\in {\mathbb C}^*$ away from a discrete set;

3.   $H^*_z (X;{\mathbb C})$ vanishes for at least one $z \in {\mathbb C}^*$.

Observe that the complexes form a holomorphic family of elliptic complexes on ${\mathbb C}^*$; therefore, exactness of at one point is equivalent to exactness away from a discrete set.  The statement now follows from the preceding discussion. 

\[R:fred\] The usual $L^2$ completion of the de Rham complex on $M$, that is, the complex $\Omega^*_0 (M)$, is not Fredholm because $H^0_z (X;{\mathbb C})$ is not zero when $z = 1$.

Finite dimensionality
---------------------

Fix a finite cell complex structure on $X$, lift it to $\tilde X$, and consider the chain complex $C_* (\tilde X,{\mathbb C})$ and its homology $H_* (\tilde X;{\mathbb C})$. The group of integers acts on both by covering translations making them into finitely generated modules over ${\mathbb C}[t,t^{-1}]$. The twisted de Rham theorem tells us that the cohomology $H^*_z (X;{\mathbb C})$ of complex is isomorphic to the cohomology of the complex ${\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(C_* (\tilde X,{\mathbb C}), {\mathbb C}_z)$, where ${\mathbb C}_z$ denotes a copy of ${\mathbb C}$ viewed as a ${\mathbb C}[t,t^{-1}]$ module with $p(t)$ acting via multiplication by $p(z)$.

\[P:finite\] Let $M$ be a smooth Riemannian manifold with a periodic end modeled on $\tilde X$. Then the following two conditions are equivalent:

1.   $H_*(M;{\mathbb C})$ is a finite dimensional vector space;

2.   $H^*_z (X;{\mathbb C})$ vanishes for at least one $z \in {\mathbb C}^*$.

It is immediate from the Mayer–Vietoris principle that $H^*(M;{\mathbb C})$ is finite dimensional if and only if $H^*(\tilde X;{\mathbb C})$ is finite dimensional.  Since $H_*(\tilde X;{\mathbb C})$ is a finitely generated module over the principal ideal domain ${\mathbb C}[t,t^{-1}]$, it admits a primary decomposition $$\label{E:prime}
H_*(\tilde X;{\mathbb C})\,=\,{\mathbb C}[t,t^{-1}]^{\ell}\,\oplus\,{\mathbb C}[t,t^{-1}]/(p_1)\,\oplus\ldots\oplus\,{\mathbb C}[t,t^{-1}]/(p_m),$$ therefore, $H^*(\tilde X;{\mathbb C})$ is a finite dimensional vector space if and only if $\ell = 0$ in this decomposition.  According to the universal coefficient theorem, $$H^*_z (X;{\mathbb C})\;=\;{\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(H_* (\tilde X;{\mathbb C}),{\mathbb C}_z)\,\oplus\,{\operatorname{Ext}}_{\,{\mathbb C}[t,t^{-1}]}\;(H_* (\tilde X;{\mathbb C}),{\mathbb C}_z),$$ hence vanishing of $H^*_z (X;{\mathbb C})$ for at least one $z$ implies that $\ell = 0$. On the other hand, an easy calculation shows that $${\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(V,{\mathbb C}_z)\;=\;{\operatorname{Ext}}_{\,{\mathbb C}[t,t^{-1}]}\;(V,{\mathbb C}_z)\;=\;0$$ for any module $V = {\mathbb C}[t,t^{-1}]/(p)$ such that $p(z) \neq 0$. Therefore, $\ell = 0$ implies that $H^*_z (X;{\mathbb C})$ must vanish for all $z$ away from the roots of the polynomials $p_1,\ldots,p_m$.

Proof of Theorem \[T:one\]
--------------------------

It follows from Propositions \[P:fred\] and \[P:finite\] that, if $H^* (\tilde X;{\mathbb C})$ is finite dimensional, the complex $\Omega^*_{\delta} (M)$ is Fredholm for all $\delta$ away from a discrete set.  To finish the proof of Theorem \[T:one\] we just need to identify this discrete set.  According to Proposition \[P:taubes\], it consists of $\delta = \ln |z|$, where $z \in {\mathbb C}^*$ are the complex numbers for which $H^*_z (X;{\mathbb C})$ fails to be zero.  To find them, note that the free part in the prime decomposition vanishes making $H_*(\tilde X;{\mathbb C})$ into a torsion module, $$\label{E:torsion}
H_*(\tilde X;{\mathbb C})\,=\,{\mathbb C}[t,t^{-1}]/(p_1)\,\oplus\ldots\oplus\,{\mathbb C}[t,t^{-1}]/(p_m).$$ According to Milnor [@milnor Assertion 4], the order ideal $(p_1\cdots p_m)$ of this module is spanned by the characteristic polynomial of $\tau_*: H_* (\tilde X;{\mathbb C}) \to H_* (\tilde X;{\mathbb C})$. The calculation with the universal coefficient theorem as in the proof of Proposition \[P:finite\] now completes the proof. 

A sufficient condition
----------------------

Let $M$ be a smooth orientable manifold with a periodic end modeled on $\tilde X$. Vanishing of $\chi (X)$ is obviously a necessary condition for the vector space $H_* (M;{\mathbb C})$ to be finite dimensional.  To come up with a sufficient condition, observe that the derivative $df$ defines a closed 1-form on $X$, and let $\xi = [df] \in H^1 (X;{\mathbb C})$ be its cohomology class.  The cup product with $\xi$ gives rise to the chain complex $$\label{E:xi}
H^0 (X;{\mathbb C}) \xrightarrow{\;\cup\,\xi\;} H^1 (X;{\mathbb C}) \xrightarrow{\;\cup\,\xi\;}\;\ldots\;\xrightarrow{\;\cup\,\xi\;} H^n (X;{\mathbb C}).$$

Suppose the chain complex is exact.  Then $H_*(M;{\mathbb C})$ is a finite dimensional vector space for any smooth orientable manifold with periodic end modeled on $\tilde X$.

This proposition can be derived as a special case of Taubes [@taubes:periodic Theorem 3.1].  That proof is analytic in nature; here is another proof which is purely topological. 

According to Proposition \[P:finite\], it is sufficient to prove that the twisted cohomology $H^*_z (X;{\mathbb C})$ vanishes for at least one $z \in {\mathbb C}^*$. We will show that it does so for all $z \neq 1$ in a sufficiently small neighborhood of 1.  For such $z$, there is a spectral sequence $(E^*_r,d_r)$ which starts at $E^*_1 = H^*(X;{\mathbb C})$, converges to $H^*_z (X;{\mathbb C})$, and whose differentials are given by the Massey products with the class $\xi$; see Farber [@farber Section 10.9].  The convergence of this spectral sequence to zero is therefore a necessary and sufficient condition for the finite dimensionality of $H_*(M;{\mathbb C})$. The chain complex is the term $(E^*_1,d_1)$ of that spectral sequence, hence its exactness is sufficient for the finite dimensionality of $H^*(X;{\mathbb C})$.

Note that the vanishing of the Euler characteristic of $X$ is not a sufficient condition for $H_*(M;{\mathbb C})$ to be finite dimensional.  An example is provided by the connected sum of $S^1 \times S^{n-1}$ with any manifold that is not a rational homology sphere but has Euler characteristic $2$.

The index calculation {#S:index}
=====================

Let $M$ be a smooth Riemannian manifold with periodic end modeled on the infinite cyclic cover $\tilde X$, and assume that $H_* (M;{\mathbb C})$ is finite dimensional.  Let $\tau: \tilde X \to \tilde X$ be a covering translation, and denote by $A_k (t)$ the characteristic polynomial of $\tau_*: H_k(\tilde X;{\mathbb C}) \to H_k(\tilde X;{\mathbb C})$ (the polynomial $A_1 (t)$ is traditionally referred to as the Alexander polynomial of the fundamental group of $X$).  Denote by $\Delta$ the set of all $\delta$ of the form $\delta = \ln|\lambda|$, where $\lambda$ is a root of the product polynomial $A_0 (t)\cdot\ldots\cdot A_{n-1}(t)$.

According to Theorem \[T:one\], the complex $\Omega^*_{\delta} (M)$ is Fredholm for all $\delta$ away from $\Delta$. Its index ${\operatorname{ind}}_{\delta}(M)$ is a piecewise constant function away from $\Delta$, where it may jump.  We wish to calculate the size of these jumps. 

Excision principle {#S:excision}
------------------

Let $\delta_1$ and $\delta_2$ be two weights in ${\mathbb R}- \Delta$ and complete the de Rham complex of $\tilde X$ in the $L^2$ norm with respect to the measure $e^{\delta_1 f(x)}\,dx$ on the negative end of $\tilde X$, and $e^{\delta_2 f(x)}\,dx$ on the positive end.  This complex will be denoted by $\Omega^*_{\delta_1\delta_2} (\tilde X)$. This is a Fredholm complex whose index will be denoted by ${\operatorname{ind}}_{\delta_1\delta_2}(\tilde X)$.

\[P:change\]${\operatorname{ind}}_{\delta_2}(M) - {\operatorname{ind}}_{\delta_1}(M)\;=\;{\operatorname{ind}}_{\delta_1\delta_2}(\tilde X)$.

Let $c\in {\mathbb R}$ be a regular value of $f: \tilde X \to {\mathbb R}$ then $Y = f^{-1} (c)$ is a submanifold of $\tilde X$ separating it as $\tilde X = \tilde X_-\,\cup\,\tilde X_+$. Write $M = Z\,\cup\,\tilde X_+$ for some smooth compact manifold $Z$ with boundary $Y$. An application of the exicision principle to these two splittings yields $${\operatorname{ind}}_{\delta_1} (M)\,+\,{\operatorname{ind}}_{\delta_1\delta_2} (\tilde X)\,=\,{\operatorname{ind}}_{\delta_2} (M)\,+\,{\operatorname{ind}}_{\delta_1\delta_1} (\tilde X).$$ Note that the complex $\Omega^*_{\delta_1\delta_1}(\tilde X)$ is exact because the complexes obtained from it by the Fourier–Laplace transform are exact for all $z$ with $|z| = e^{\delta_1}$. Therefore, ${\operatorname{ind}}_{\delta_1\delta_1} (\tilde X) = 0$ and the proof is complete. 

Computing cohomology of $\Omega^*_{\delta_1\delta_2}(\tilde X$)
---------------------------------------------------------------

We will proceed by several reductions, the first being from weighted forms to weighted cellular cochains.  To be precise, fix a finite cell complex structure on $X$ and lift it to $\tilde X$. Also, introduce the Hilbert space $\ell^2_{\delta_1\delta_2}$ of the sequences $\{x_k\;|\; k\in \mathbb Z\}$ of complex numbers such that $$\sum_{k < 0}\; e^{2\delta_1 k}\,|x_k|^2\; < \;\infty\qquad\text{and}
\qquad \sum_{k > 0}\; e^{2\delta_2 k}\,|x_k|^2\; < \;\infty.$$

Theorem 2.17 of Miller [@miller], which is a weighted version of the $L^2$ de Rham theorem, see [@dodziuk; @lueck:L2], establishes an isomorphism between the cohomology of $\Omega^*_{\delta_1\delta_2} (\tilde X)$ and the cellular cohomology of $\tilde X$ with $\ell^2_{\delta_1\delta_2}$ coefficients.  Miller actually uses weighted [*simplicial*]{} cohomology, but this is readily seen to be isomorphic to the more standard and convenient cellular version. 

\[P:shift\] View $\ell^2_{\delta_1\delta_2}$ as a ${\mathbb C}[t,t^{-1}]$ module with $t$ acting as the right shift operator, $t(x_k) = x_{k+1}$. Then for all but finitely many $\delta_1$ and $\delta_2$, the cohomology of $\tilde X$ with $\ell^2_{\delta_1\delta_2}$ coefficients equals the homology of the complex $$\label{E:hom-l2}
{\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(C_*(\tilde X,{\mathbb C}),\,\ell^2_{\delta_1\delta_2}).$$

This will follow as soon as we show that the images of the boundary operators ${\partial}$ in complex are closed.  These boundary operators $${\partial}: \left(\ell^2_{\delta_1 \delta_2}\right)^k \to \left(\ell^2_{\delta_1 \delta_2}\right)^{\ell}$$ are matrices whose entries are Laurent polynomials in $t$. Since ${\mathbb C}[t,t^{-1}]$ is a principal ideal domain, each ${\partial}$ will have a diagonal matrix in properly chosen bases.  The statement now follows from the fact that the operator $t - \lambda: \ell^2_{\delta_1 \delta_2} \to \ell^2_{\delta_1 \delta_2}$ is Fredholm for all $\lambda$ with $|\lambda|$ different from $e^{\delta_1}$ and $e^{\delta_2}$; see for instance Conway [@conway Proposition 27.7 (c)]. 

The universal coefficient theorem now tells us that the $\ell^2_{\delta_1 \delta_2}$ cohomology of $\tilde X$ is isomorphic to $$\label{E:coeff}
{\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(H_*(\tilde X;{\mathbb C}),\ell^2_{\delta_1 \delta_2})\,\oplus\,
{\operatorname{Ext}}_{\,{\mathbb C}[t,t^{-1}]}\;(H_*(\tilde X;{\mathbb C}),\ell^2_{\delta_1 \delta_2}).$$ Recall from the proof of Theorem \[T:one\] that $H_* (\tilde X;{\mathbb C})$ is a torsion module whose order ideal is spanned by the characteristic polynomial of $\tau_*: H_* (\tilde X;{\mathbb C}) \to H_* (\tilde X;{\mathbb C})$. Therefore, our next step will be to compute , one cyclic module at a time. 

Let $\lambda$ be a complex number such that $|\lambda|$ is different from $e^{\delta_1}$ and $e^{\delta_2}$. Then, for any cyclic module $V = {\mathbb C}[t,t^{-1}]/(t-\lambda)^m$, we have $${\operatorname{Ext}}_{\,{\mathbb C}[t,t^{-1}]}\;(V,\ell^2_{\delta_1 \delta_2})\,=\,0.$$

We already know from the proof of Proposition \[P:shift\] that the operator $t - \lambda: \ell^2_{\delta_1 \delta_2} \to \ell^2_{\delta_1 \delta_2}$ is Fredholm.  In addition, one can easily check that all finite sequences belong to its image.  Since such sequences are dense in $\ell^2_{\delta_1 \delta_2}$, the operator $t - \lambda$ is surjective, and so are the operators $(t - \lambda)^m$ for all $m$. The result is now immediate from the definition of ${\operatorname{Ext}}$.

Let $\lambda$ be a complex number such that $|\lambda|$ is different from $e^{\delta_1}$ and $e^{\delta_2}$. Assume that $\delta_2 < \delta_1$. Then, for any cyclic module $V = {\mathbb C}[t,t^{-1}]/(t-\lambda)^m$, the dimension of ${\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\;(V,\ell^2_{\delta_1 \delta_2})$ is $m$ if $e^{\delta_2} < |\lambda| < e^{\delta_1}$, and zero otherwise. 

For such a module, ${\operatorname{Hom}}_{\,{\mathbb C}[t,t^{-1}]}\,(V,\ell^2_{\delta_1 \delta_2})$ equals the kernel of the operator $(t - \lambda)^m: \ell^2_{\delta_1 \delta_2} \to \ell^2_{\delta_1 \delta_2}$. Computing this kernel is a straightforward exercise with infinite series. 

Proof of Theorem \[T:two\]
--------------------------

Let $\lambda$ be a root of the product polynomial $A_0(t)\cdots A_{n-1}(t)$ of multiplicity $m = m_0 + \ldots + m_{n-1}$, where $m_k$ is the multiplicity of $\lambda$ as a root of $A_k(t)$. Choose generic $\delta_1$ and $\delta_2$ so that $e^{\delta_2} < |\lambda| < e^{\delta_1}$ and there are no other roots of $A_0(t)\cdots A_{n-1}(t)$ whose absolute values fit in this interval.  It follows from Proposition \[P:change\] and the cohomology calculation in the previous section that $${\operatorname{ind}}_{\delta_1} (M)\;=\;{\operatorname{ind}}_{\delta_2} (M)\;-\;\sum\;(-1)^k\,m_k,$$ which is exactly the formula claimed in Theorem \[T:two\]. 

Discussion and examples
=======================

Let $M$ be a smooth Riemannian manifold of dimension $n$ with a periodic end modeled on $\tilde X$ and suppose that $H^*(\tilde X;{\mathbb C})$ is finite dimensional.  Then, for any $\delta \in {\mathbb R}- \Delta$, the de Rham complex $\Omega^*_{\delta} (M)$ is Fredholm and its index is given by the formula $${\operatorname{ind}}_{\delta} (M)\;=\; (-1)^n\,\chi(M)\; + \;\sum\;(-1)^k\;\# \{\lambda\,|\, A_k(\lambda)=0,\;|\lambda| > e^{\delta}\,\},$$ where the roots $\lambda$ of $A_k (t)$ are counted with their multiplicities.  This formula is obtained by combining Theorem \[T:two\] with Miller’s theorem [@miller] that ${\operatorname{ind}}_{\delta} (M) = (-1)^n\,\chi (M)$ for sufficiently large $\delta > 0$. Miller [@miller] also shows that the function ${\operatorname{ind}}_{\delta} (M)$ is even or odd depending on whether $n$ is even or odd.  This is consistent with the above formula because of Blanchfield duality which says that $A_k(\lambda) = 0$ if and only if $A_{n-k-1} (1/\lambda) = 0$ with matching multiplicities. 

A manifold with product end is a smooth Riemannian manifold whose end is modeled on $\tilde X = \mathbb R \times Y$, where $Y$ is a closed Riemannian manifold.  The metric on ${\mathbb R}\times Y$ is presumed to be the product metric.  The index theory on such manifolds has been studied by Atiyah, Patodi and Singer [@aps:I].  The covering translation induces an identity map $\tau_*$ on the homology of ${\mathbb R}\times Y$. Since $\lambda = 1$ is the only root of the characterictic polynomial of $\tau_*$ the complex $\Omega^*_{\delta}\,(M)$ is Fredholm for all $\delta \ne 0$. Its index ${\operatorname{ind}}_{\delta} (M)$ equals $\chi (M)$ if the dimension of $M$ is even, and ${\operatorname{sign}}(-\delta)\cdot \chi(M)$ if the dimension of $M$ is odd.  Note that the same is true for any manifold whose periodic end is modeled on $\tilde X$ such that the characteristic polynomial of $\tau_*: H_* (\tilde X;{\mathbb C}) \to H_* (\tilde X;{\mathbb C})$ only has unitary roots. 

This example originates in Fox’s ‘Quick Trip’ [@fox:trip Example 11].  Fox constructs a 2-knot in the 4-sphere with with the property that the infinite cyclic cover of its exterior has first homology isomorphic to the additive group of dyadic rationals.  A nice plumbing construction of this knot described in Rolfsen’s book [@rolfsen:knots Section 7.F] shows that it has a Seifert surface diffeomorphic to $S^1 \times S^2 - D^3$. Perform a surgery on the knot so that the Seifert surface is capped off by the core 3-disk of the surgery.  The resulting manifold $X$ has integral homology of $S^1 \times S^3$. It follows from a calculation in [@rolfsen:knots Section 7.F] that the characteristic polynomials $A_k (t)$ of the covering translation $\tau_*: H_k (\tilde X;{\mathbb C}) \to H_k (\tilde X;{\mathbb C})$ are as follows: $A_0 (t) = t - 1$, $A_1 (t) = t - 2$, $A_2 (t) = t - 1/2$, and $A_3 (t) = t - 1$. Cut $\tilde X$ along a copy of $S^1 \times S^2$ and fill it in by $D^2 \times S^2$ to obtain an end-periodic manifold $M$. A straightforward calculation shows that $\chi (M) = 2$. The complex $\Omega^*_{\delta}(M)$ is Fredholm away from $\delta = 0$ and $\delta = \pm \ln 2$. Its index is equal to 1 if $0 < |\delta| < \ln 2$, and is equal to 2 otherwise. 

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