Regular Cell Complex

Spectral Sheaf Theory


Regular cell complexes are a type of cell complexes that are particularly instrumental in TDA and spectral sheaf theory. Regular cell complexes provide combinatorial and topological insights into various spaces, including simplicial and cubical complexes.


A regular cell complex is a topological space $X$ partitioned into subspaces $\{X_\alpha\}_{\alpha\in PX}$ satisfying the following conditions:

  1. For each $x \in X$, every sufficiently small neighborhood of $x$ intersects finitely many $X_\alpha$.
  2. For all $\alpha, \beta$, $X_\alpha \cap X_\beta \neq \emptyset$ only if $X_\beta \subseteq X_\alpha$.
  3. Every $X_\alpha$ is homeomorphic to $\mathbb{R}^{n_\alpha}$ for some $n_\alpha$.
  4. For every $\alpha$, there is a homeomorphism of a closed ball in $\mathbb{R}^{n_\alpha}$ to $X_\alpha$ that maps the interior of the ball homeomorphically onto $X_\alpha$.

Condition (2) implies that the set $PX$ has a poset structure, given by $\beta \leq \alpha$ if $X_\beta \subseteq X_\alpha$. This is known as the face poset of $X$. The regularity condition (4) implies that all topological information about $X$ is encoded in the poset structure of $PX$. For our purposes, we will identify a regular cell complex with its face poset, writing the incidence relation $\beta \prec \alpha$. The class of posets that face incidence posets that arises from a continuous map between their associated topological spaces. In particular, morphisms of simplicial and cubical complexes will qualify as morphisms of regular cell complexes.


Main article: Spectral Sheaf Theory, Hodge Laplacian

The class of regular cell complexes includes simplicial complexes, cubical complexes, and so-called multigraphs (as 1-dimensional cell complexes). As nearly every space that can be characterized combinatorially can be represented as a regular cell complex, these will serve well as a default class of spaces over which a spectral sheaf theory|spectral theory of sheaves can be developed. In practice, spectral theory of complexes has been largely restricted to the study of simplicial complexes. A number of results about the spectra of Hodge Laplacians of regular cell complexes can be obtained restricting to the constant sheaf.

See also


Topics in Topological Data Analysis

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