Hirzebruch's signature formula is not valid for manifolds with boundary. An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely: $$sign (M)=L(M)[M]+\eta(\partial M)$$

Yet better,in some situations,this invariant could be used to distinguish homeo types within a simple homotopy type.

My question is about basic properties of eta invariant:Is there a formula for

$\bullet$ $\eta(M\times N)$ where $M$ and $N$ are closed manifolds

$\bullet$ $\eta(\widetilde{M})$ where $\widetilde{M}$ is a regular covering space of closed manifold $M$

$\bullet$ $\eta(2M)$ where $M$ is a manifold with boundary and $2M$ is the double $M\cup_{\partial M} M$.

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    $\begingroup$ Regarding $\eta(\widetilde{M})$: the eta invariant on the sphere is zero, but there are lens spaces with nonzero eta invariants. So there is not likely to be a simple formula. $\endgroup$ Sep 19 '15 at 17:52
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    $\begingroup$ Regarding $\eta(2M)$: the standard eta invariant is only defined on closed manifolds; to define a counterpart for manifolds with boundary you would need to explore the index defect for manifolds with corners. There is literature on this (I don't have references on hand) and people have looked at what happens to the eta invariant when you glue, but I recall the answers being a bit complicated. $\endgroup$ Sep 19 '15 at 17:57

I don't have a reference for all of these, but the answers can be worked out from the APS theorem. Note that you should be a little careful with the notation; these are really metric invariants, and one usually works with metrics that are products near the boundary.

For the first, to discuss $\eta(M \times N)$ then $M \times N$ is supposed to be an odd-dimensional closed manifold of dimension $4k-1$, so let's suppose $M$ has odd dimension and $N$ has even dimension. I claim that $\eta(M \times N) = \eta(M) \sigma(N)$ if dim(M) is of the form $4j-1$. The answer is $0$ if dim(M) is of the form $4j+1$.

Let's assume also that $M$ is an oriented boundary, say $M = \partial W$. (In general, you know that some number, say $n$ of copies of $M$ is a boundary, and you apply the argument those $n$ copies.) Then $M \times N= \partial(W \times N)$ and you can use the multiplicativity of the L-class and the signature (this is true for manifolds with boundary) plus (or should I say times!) the signature theorem applied to $N$ to get the result. The other case is easier, since the signature of $N$ is $0$ and the L-class of the product is 0 for simple dimensional reasons.

For the second, the answer is definitely no. Indeed, much of the second APS paper is devoted to the $\rho$ invariant, which roughly speaking measures the difference between $\eta(\tilde{M})$ and (degree of the covering times)$\eta(M)$. The $\rho$ invariant is in an appropriate sense a topological invariant; perhaps this is what you are referring to in reference to distinguishing homeomorphisms types.

For the third, $\eta(2M) = 0$ since $2M$ (with a metric obtained by gluing together identical metrics on the two copies of $M$) has an orientation reversing isometry. This point is discussed in the introduction to the first APS paper.

  • $\begingroup$ Could you please give an indication as for where the first (product) property comes from? $\endgroup$ Aug 6 '18 at 15:18

You may find some information from the following paper:

Donnelly, Harold

Eta invariant of a fibered manifold.

Topology 15 (1976), no. 3, 247–252.

Review from Mathscinet


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