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**The cell-like approximation theorem in dimension 5.**
*(English)*
Zbl 1133.57013

Authors’ summary: The cell-like approximation theorem of R. D. Edwards characterizes the \(n\)-manifolds precisely as the resolvable ENR homology \(n\)-manifolds with the disjoint disks property for \(5\leq n<\infty\). Since no proof for the \(n=5\) case has ever been published, we provide the missing details about the proof of the cell-like approximation theorem in dimension 5.

The cell-like approximation theorem of Robert D. Edwards [Proc. Int. Congr. Math. (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 111–127 (1980; Zbl 0428.57004)] states that a cell-like mapping \(F:M\rightarrow X\) from an \(n\)-manifold \(M\), \(n\geq 5\), onto a metric space \(X\) can be approximated by homeomorphisms if and only if \(X\) is finite-dimensional and has the disjoint disks property (DDP). A fundamentally important corollary is the following characterization of topological manifolds: a (metric) space \(X\) is an \(n\)-manifold \(n\geq 5\), if and only if \(X\) is a resolvable ENR homology \(n\)-manifold having the disjoint disks property. Edwards outlined the proof in the above cited work, and a fairly complete argument was given by the first author of the present paper in 2007 [Decompositions of manifolds, AMS Chelsea, Providence, RI, (2007; Zbl 1130.57001)]; both focused on the case \(n> 5\). This paper presents an argument concerning the special case \(n=5\), a proof for which has never been published, and details of which are not widely known.

To achieve its main purpose, the paper contributes to decomposition theory by providing conditions under which a cell-like decomposition of a \(5\)-manifold can be shrunk out while keeping certain special \(2\)-complexes fixed. Equivalently, it provides conditions under which one cell-like map of a \(5\)-manifold to itself can be replaced with another such cell-like map that is the identity on the 2-complex. The argument retraces most of the steps required to establish Edwards’ cell-like approximation theorem. The authors prove the following theorem: Theorem 6.1. (Main Theorem). Suppose \(F:M\rightarrow X\) is a cell-like map defined on a 5-manifold \(M\) where \(\dim X=5\) and \(X\) has the DDP. Then \(F\) can be approximated by a cell-like map \(\Psi:M\rightarrow X\) such that the embedding dimension of the nondegeneracy set \(N_\Psi\) is at most 2.

The procedure for proving the Main Theorem is outlined as follows: Step 1. The improvement of the map \(F:M\rightarrow X\) so that \(F\) is 1-1 over a “very large” countable dense collection of LCC\(^1\) embedded 2-complexes in \(X\). The lifts of the maps in the collection will serve as targets for moves in the next step. Now let \(F:M\rightarrow X\) denote the improved map and \(G\) denote the induced decomposition.

Step 2. Establishing a procedure to move a 2-complex \(K\) in \(M\) ”near” a target. In particular, the authors show that \(K\) may be almost entirely pushed onto a target by an isotopy, the deficit being realized on a finite set of disks in \(K\) of arbitrarily small size and \(K\) being mapped into \(G_{<\varepsilon}\) (the union of all elements of \(G\) whose diameter is less than \(\varepsilon \)) for some prescribed \(\varepsilon\).

Step 3. Determining an ambient pseudo-isotopy of \(M\), the limit of a composition of controlled pushes, the covers of the isotopies described in Step 2, such that the end of the ambient pseudo-isotopy \(h\) satisfies: (a) \(h\) is 1-1 on \(K\) and takes \(K\) off \(N_G\), (b) \(G_h=\{h^{-1}(x)\}\), the decomposition induced by \(h\), has nondegeneracy set \(N_h\) such that \(dem(N_h)\leq2\).

Step 4. Applying shrinking fixing a 2-complex to obtain a self-homeomorphism \(\Phi:M\rightarrow M\) such that: (a) \(\Phi|\lceil K\rfloor\), (b) \(\Phi=h\) outside \(h^{-1}(V)\) where \(V\) is a small neighborhood of \(h(K)\), (c) \(\varrho(\Phi,h)<\varepsilon\).

Step 5. Giving the 2-skeleta of a sequence \(\{K_i\}\) of triangulations whose mesh tends to zero, to apply Steps 3 and 4 to determine homeomorphisms \(\Phi_i:M\rightarrow M\), removing \(K_i\) from the approximate nondegeneracy set, and proper cell-like maps \(F_i=F_{i-1}\Phi_i:M\rightarrow X\) (\(F_0=F\)) such that the limit map \(\Psi\) of \(\{F_i\}\) is 1-1 over the image of the 2-skeleta. Then \(\Psi\) will be an approximation of \(F\) such that \(N_\Psi\) has embedding dimension at most 2, the desired result.

The cell-like approximation theorem of Robert D. Edwards [Proc. Int. Congr. Math. (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 111–127 (1980; Zbl 0428.57004)] states that a cell-like mapping \(F:M\rightarrow X\) from an \(n\)-manifold \(M\), \(n\geq 5\), onto a metric space \(X\) can be approximated by homeomorphisms if and only if \(X\) is finite-dimensional and has the disjoint disks property (DDP). A fundamentally important corollary is the following characterization of topological manifolds: a (metric) space \(X\) is an \(n\)-manifold \(n\geq 5\), if and only if \(X\) is a resolvable ENR homology \(n\)-manifold having the disjoint disks property. Edwards outlined the proof in the above cited work, and a fairly complete argument was given by the first author of the present paper in 2007 [Decompositions of manifolds, AMS Chelsea, Providence, RI, (2007; Zbl 1130.57001)]; both focused on the case \(n> 5\). This paper presents an argument concerning the special case \(n=5\), a proof for which has never been published, and details of which are not widely known.

To achieve its main purpose, the paper contributes to decomposition theory by providing conditions under which a cell-like decomposition of a \(5\)-manifold can be shrunk out while keeping certain special \(2\)-complexes fixed. Equivalently, it provides conditions under which one cell-like map of a \(5\)-manifold to itself can be replaced with another such cell-like map that is the identity on the 2-complex. The argument retraces most of the steps required to establish Edwards’ cell-like approximation theorem. The authors prove the following theorem: Theorem 6.1. (Main Theorem). Suppose \(F:M\rightarrow X\) is a cell-like map defined on a 5-manifold \(M\) where \(\dim X=5\) and \(X\) has the DDP. Then \(F\) can be approximated by a cell-like map \(\Psi:M\rightarrow X\) such that the embedding dimension of the nondegeneracy set \(N_\Psi\) is at most 2.

The procedure for proving the Main Theorem is outlined as follows: Step 1. The improvement of the map \(F:M\rightarrow X\) so that \(F\) is 1-1 over a “very large” countable dense collection of LCC\(^1\) embedded 2-complexes in \(X\). The lifts of the maps in the collection will serve as targets for moves in the next step. Now let \(F:M\rightarrow X\) denote the improved map and \(G\) denote the induced decomposition.

Step 2. Establishing a procedure to move a 2-complex \(K\) in \(M\) ”near” a target. In particular, the authors show that \(K\) may be almost entirely pushed onto a target by an isotopy, the deficit being realized on a finite set of disks in \(K\) of arbitrarily small size and \(K\) being mapped into \(G_{<\varepsilon}\) (the union of all elements of \(G\) whose diameter is less than \(\varepsilon \)) for some prescribed \(\varepsilon\).

Step 3. Determining an ambient pseudo-isotopy of \(M\), the limit of a composition of controlled pushes, the covers of the isotopies described in Step 2, such that the end of the ambient pseudo-isotopy \(h\) satisfies: (a) \(h\) is 1-1 on \(K\) and takes \(K\) off \(N_G\), (b) \(G_h=\{h^{-1}(x)\}\), the decomposition induced by \(h\), has nondegeneracy set \(N_h\) such that \(dem(N_h)\leq2\).

Step 4. Applying shrinking fixing a 2-complex to obtain a self-homeomorphism \(\Phi:M\rightarrow M\) such that: (a) \(\Phi|\lceil K\rfloor\), (b) \(\Phi=h\) outside \(h^{-1}(V)\) where \(V\) is a small neighborhood of \(h(K)\), (c) \(\varrho(\Phi,h)<\varepsilon\).

Step 5. Giving the 2-skeleta of a sequence \(\{K_i\}\) of triangulations whose mesh tends to zero, to apply Steps 3 and 4 to determine homeomorphisms \(\Phi_i:M\rightarrow M\), removing \(K_i\) from the approximate nondegeneracy set, and proper cell-like maps \(F_i=F_{i-1}\Phi_i:M\rightarrow X\) (\(F_0=F\)) such that the limit map \(\Psi\) of \(\{F_i\}\) is 1-1 over the image of the 2-skeleta. Then \(\Psi\) will be an approximation of \(F\) such that \(N_\Psi\) has embedding dimension at most 2, the desired result.

Reviewer: Ioan Pop (Iaşi)