*I have asked [the same question on math.SE](https://math.stackexchange.com/questions/4620954/is-the-n-2-th-heat-kernel-coefficient-topological), without much success so I'm trying my luck here too.*


Let $M$ be an $n$-dimensional manifold, with $n$ *even* and consider the heat kernel of the Laplacian on $M$:
$$K\_M(t) := \mathrm{Tr}\Big(\mathrm{e}^{-t \Delta\_M}\Big) \equiv \sum\_{\lambda\in\mathrm{spec}(\Delta\_M)} \mathrm{e}^{-t \lambda}.$$
The heat kernel admits an asympotic, small-$t$ expansion as
$$K\_M(t) = \frac{1}{(4\pi t)^{n/2}}\sum\_{k=0}^\infty b\_k^{(M)} t^k,$$
where $b\_k^{(M)}$ are known as the *heat kernel coefficients* and are given by integrals of local geometric data of $M$.


I am particularly interested in what I'll call *middle-dimensional heat kernel coefficient*, namely
$$ b\_{n/2}^{(M)} = b\_{\mathrm{dim}M/2}^{(M)}.$$
This coefficient is interesting because it is very closely related to the value of the Minakshisundaram-Pleijel zeta function (i.e. the spectral zeta function for the Laplacian) at zero.


Is an explicit formula for the calculation of $b\_{n/2}^{(M)}$ known? In particular, is it a topological invariant? It seems to me that it wants to be related to the number of connected components of $M$ or to some (alternating?) sum of Betti numbers of $M$, but I haven't managed to show it.


If anyone feels brave, I'd be curious about how it generalizes to $p$-forms; namely what's the middle-dimensional heat kernel coefficient associated to the Laplacian, $\Delta\_p = \mathrm{d}\delta + \delta \mathrm{d}$, acting on $p$-forms on $M$?



Is the $n/2$-th heat kernel coefficient topological?
For $n=2$, the answer is given by $\frac{E}{6}$, where $E$ is the Euler characteristic of $M$, see McKean Jr, H. P., & Singer, I. M. (1967). Curvature and the eigenvalues of the Laplacian. Journal of Differential Geometry, 1(1-2), 43-69.


For $n>2$, if it is topological, then it is none of what you conjectured. Consider the case of hyperbolic manifolds: since those are locally isometric to the hyperbolic space, the coefficient $b^{M}\_{n/2}$ is proportional to the hyperbolic volume (Which, incidentally, is indeed topological by Mostow rigidity.)


**Update:** It is not topological for $n>2$. Let $M\_1=\mathbb{T}^2\times\mathbb{S}^2$ and $M\_2=(2\mathbb{T}^2)\times \mathbb{S}^2,$ where $2\mathbb{T}^2$ denotes the torus scaled by the factor of $2$. Then, $M\_1$ and $M\_2$ are homeomorphic and $b\_2^{M\_2}=4b\_2^{M\_1},$ because both spaces are homogeneous and locally isometric to each other. On the other hand, $$P^{M\_1}\_t(x,x)=P^{\mathbb{T}^2}\_t(x,x)P^{\mathbb{S}^2}\_t(x,x)=\left(\frac{1}{4\pi t}+o(t^{100})\right)P^{\mathbb{S}^2}\_t(x,x).$$ Therefore, we only need to show that $b^{\mathbb{S}^2}\_2$ does not vanish. According to the table on page 63 of the above reference, this is indeed the case.


