The Reeb Space

de Silva, Munch, and Patel (2015) observed that \(\gamma \circ \lambda\) is closely related to another endofunctor on topological spaces over \(M\), the Reeb space.

  • Definition. Given a continuous map \(f \colon X \rightarrow M\) and \(x \in X\) let \(\pi_f (x)\) be the connected component of \(x\) in \(f^{-1}(f(x))\). In this way we obtain a function \(\pi_f \colon X \rightarrow 2^X\) and we endow \(\pi_f (X)\) with the quotient topology[1]. By the universal property of the quotient space there is a unique continuous function \(\tilde{f} \colon \pi_f (X) \rightarrow M\) such that \(\tilde{f} \circ \pi_f = f\) and we define \(\rho_f = \tilde{f}\).

With this definition \(\rho\) forms an endofunctor on topological spaces over \(M\) and \(\pi\) a natural transformation from \({\operatorname{id}}\) to \(\rho\). Given a continuous map \(f \colon X \rightarrow M\) for a locally connected topological space \(X\) the universal property of the quotient space induces a unique map \(\phi_f \colon \pi_f (X) \rightarrow {\operatorname{dis}}\lambda_f\) such that \(\phi_f \circ \pi_f = \eta_f\) and thus in particular \(\rho_f = (\gamma \circ \lambda)_f \circ \pi_f\), hence we have the following commutative diagram \[ \xymatrix{ & {\operatorname{id}}\ar[dl]_{\pi} \ar[dr]^{\eta} \\ \rho \ar[rr]_{\phi} & & \gamma \circ \lambda } \] in the category of endofunctors on locally connected topological spaces over \(M\).

  • Proposition. The natural transformation \(\lambda \circ \phi\) from \(\lambda \circ \rho\) to \(\lambda \circ (\gamma \circ \lambda)\) is an isomorphism.

  • Proof. We apply \(\lambda\) to the previous diagram and obtain \[ \xymatrix{ & \lambda \ar[dl]_{\lambda \circ \pi} \ar[dr]^{\lambda \circ \eta} \\ \lambda \circ \rho \ar[rr]_{\lambda \circ \phi} & & \lambda \circ \gamma \circ \lambda . } \] Since \(\lambda \circ \pi\) is an isomorphism, it suffices to show that \(\lambda \circ \eta\) is an isomorphism. Given \(f \colon X \rightarrow M\) we apply the inverse bijection induced by the adjunction \((\lambda \dashv \gamma, \eta)\) to the diagram \[ \xymatrix{ f \ar[dr]^{\eta_f} \ar[dd]_{\eta_f} \\ & (\gamma \circ \lambda)_f \\ (\gamma \circ \lambda)_f \ar[ur]_{{\operatorname{id}}} } \] and obtain \[ \xymatrix{ \lambda_f \ar[dr]^{{\operatorname{id}}} \ar[dd]_{(\lambda \circ \eta)_f} \\ & \lambda_f \\ (\lambda \circ \gamma \circ \lambda)_f , \ar[ur]_{(\varepsilon \circ \lambda)_f} } \] hence \((\lambda \circ \eta)_f\) is the inverse to \((\varepsilon \circ \lambda)_f\).

  1. Corollary. \(\phi\) and \(\eta \circ \rho\) are naturally ismorphic as functors from the category of topological spaces over \({\mathbb{R}}\) to the category of homomorphisms in the category of topological spaces over \({\mathbb{R}}\).

  • Example. Let \(f \colon X \rightarrow {\mathbb{R}}\) be a proper Morse function, then the critical points of \(f\) are isolated and since \(f\) is proper, it’s critical values are isolated as well. Hence for each \(r \in {\mathbb{R}}\) there is an \(\varepsilon_r > 0\) such that for all \(0 < \delta \leq \varepsilon_r\) the inclusion of \(f^{-1} (r)\) into \(f^{-1} ((r - \delta, r + \delta))\) is a homotopy equivalence and thus \(\phi_f\) is a homeomorphism.

de Silva, Munch, and Patel (2015) provide a self-contained treatment of the above when \(\lambda\) and \(\gamma\) are restricted to full subcategories of topological spaces over \({\mathbb{R}}\) respectively cosheaves on \({\mathbb{R}}\). When \(\phi\) is restricted to this subcategory of topological spaces over \({\mathbb{R}}\) referred to as constructible \({\mathbb{R}}\)-spaces, then \(\phi\) is a natural isomorphism. Further the authors provide a geometric description of the resulting subcategory of cosheaf spaces over \({\mathbb{R}}\). They refer to this category as Reeb or as the category of \({\mathbb{R}}\)-graphs.



[1] This is in line with the previously made assumption, since quotient spaces of locally connected spaces are again locally connected.