# Do higher dimensional maxima of a real valued multivariable function form a cell complex?

Suppose $f: R^n \rightarrow R$ is a positive real valued function. Let $\lambda_1, \ldots , \lambda_i$ be the first $i$ ordered eigenvalues of the Hessian $Hess(f)$. Let $v_1, \ldots, v_i$ be the corresponding unit eigenvectors. Suppose $V = [v_1, \ldots, v_{n-d}]$. Define a $d-$dimensional ridge point as a point where $V^T \nabla f = 0$ and where $\lambda_1, \ldots, \lambda_{n-d} < 0$. Each point in the support of $f$ can then be classified as a ridge point. My question is, does this classification yield a cell complex? Or at least, are the $k$ dimensional ridge points always bounded by $k-1$ dimensional ridge points? This construction seems close enough to a Morse complex but it's different in that I'm only interested in the local maxima, and that two maxima need not be joined by an integral curve. Any help would be appreciated, thanks!

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Does f have continuous second derivatives? –  S. Carnahan Aug 9 '10 at 17:16
Yes, we can assume that $f$ is a $C^\infty$ function for now. –  sai Aug 9 '10 at 17:21
Maybe this should be "extrema" in general, not just maxima/minima. –  Mikola Aug 9 '10 at 17:41