Chapman’s category isomorphism for arbitrary ARs.

*(English)*Zbl 0592.57013Let Q denote the Hilbert cube, Sh(Q) the shape category restricted to the Z-sets of Q and \(Top_{wh}(Q)\) the weak proper homotopy category of complements of Z-sets in Q. A well-known theorem of T. A. Chapman [ibid. 76, 181-193 (1972; Zbl 0262.55016)] establishes a category isomorphism \(Top_{wh}(Q)\to Sh(Q)\), which associates with every object M its complement Q-M. An analogous result holds for the proper homotopy category \(Top_ h(Q)\) of the complements of Z-sets in Q and the strong shape category sSh(Q) [see D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology (Lect. Notes Math. 542) (1976; Zbl 0334.55001); Y. Kodama and J. Ono, Fundam. Math. 105, 29-39 (1979; Zbl 0425.54016)]. In the present paper the author generalizes these results to the situation where Q is replaced by an arbitrary AR W. Moreover, he establishes analogous complement theorems for homotopy categories of compact Z-sets of W, defined as unstable zero-sets.

It is well known that for the AR \(s=\prod R_ n\), \(R_ n={\mathbb{R}}\), \(n=1,2,...\), the spaces s-X and X are homeomorphic for any compactum \(X\subseteq s\). This shows that the above mentioned complement theorems do not admit direct generalizations to AR’s. The author achieves his results by recurring to uniform structures. For any \(W\in AR\) he establishes two category isomorphisms \(T_{wh}: C_{wh}(W^*)\to Sh(W)\) and \(T_ h: C_ h(W^*)\to sSh(W)\), which map objects \(M\subseteq W\) to their complements W-M. Here \(W^*\) is the universal uniformity of W (which is complete). C is the category of uniform spaces and complete maps, i.e. maps f having the property that \(f^{-1}(M)\) is complete whenever M is complete. \(C_ h(W^*)\) \((C_{wh}(W^*))\) is the homotopy (weak homotopy) category associated with C, restricted to the uniform subspaces \((W-X)^*\subseteq W^*\), where X ranges over compact Z-sets of W.

The analogous theorem for homotopy establish category isomorphisms \(R_{wh}: UC_{wh}(W^*)\to Top_{wh}(W)\) and \(R_ h: UC_ h(W^*)\to Top_ h(W)\). Here the category C is replaced by the category UC of uniform spaces and complete maps \(f: Y\to Z\) having the property that \(f| A: A\to Z\) is uniformly continuous whenever A is a closed totally bounded subspace of Y.

It is well known that for the AR \(s=\prod R_ n\), \(R_ n={\mathbb{R}}\), \(n=1,2,...\), the spaces s-X and X are homeomorphic for any compactum \(X\subseteq s\). This shows that the above mentioned complement theorems do not admit direct generalizations to AR’s. The author achieves his results by recurring to uniform structures. For any \(W\in AR\) he establishes two category isomorphisms \(T_{wh}: C_{wh}(W^*)\to Sh(W)\) and \(T_ h: C_ h(W^*)\to sSh(W)\), which map objects \(M\subseteq W\) to their complements W-M. Here \(W^*\) is the universal uniformity of W (which is complete). C is the category of uniform spaces and complete maps, i.e. maps f having the property that \(f^{-1}(M)\) is complete whenever M is complete. \(C_ h(W^*)\) \((C_{wh}(W^*))\) is the homotopy (weak homotopy) category associated with C, restricted to the uniform subspaces \((W-X)^*\subseteq W^*\), where X ranges over compact Z-sets of W.

The analogous theorem for homotopy establish category isomorphisms \(R_{wh}: UC_{wh}(W^*)\to Top_{wh}(W)\) and \(R_ h: UC_ h(W^*)\to Top_ h(W)\). Here the category C is replaced by the category UC of uniform spaces and complete maps \(f: Y\to Z\) having the property that \(f| A: A\to Z\) is uniformly continuous whenever A is a closed totally bounded subspace of Y.

Reviewer: S.Mardešić

##### MSC:

57N25 | Shapes (aspects of topological manifolds) |

55P55 | Shape theory |

54C56 | Shape theory in general topology |

54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |

54E15 | Uniform structures and generalizations |