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Finite-dimensional categorical complement theorems in strong shape theory and a principle of reversing maps between open subsets of spheres. (English) Zbl 0721.55006
This paper continues the author’s work in finding generalized shape complement theorems. In a previous paper [Compos. Math. 68, 161–173 (1988; Zbl 0665.55006)] the author proved a theorem which related the shape of a space $$X$$ embedded in an ANR $$M$$ to homotopical information about the complement of $$X$$ in $$M$$. The theorem requires restrictions on the embedding, the fundamental dimension of $$X$$, and the connectivity of $$M$$. The present paper is concerned with relating the strong shape of $$X$$ to the complement of $$X$$ embedded in an appropriate way in an appropriate ANR. The theorems obtained bear an appropriate resemblance of those obtained in his previous paper. The following duality theorem in the paper is quoted to give a flavor of the results obtained.
Theorem B. Let $$d(n)=\max \{k\mid 2k+2\leq n\}$$ and $$c(n)=\max \{k\mid 4k\leq n\}$$. Let $$T_ n$$ be the full subcategory of the homotopy category HTop whose objects are all complements $$S^ n\setminus X$$ of $$c(n)$$-shape-connected compacta $$X\subset S^ n$$ having $$\text{Fd }X\leq d(n)-1$$ and satisfying the inessential loops condition ILC, and let $$H_{d(n)}P$$ be the proper $$d(n)$$-homotopy category. There exists a contravariant full embedding $$\Delta : T_ n\to H_{d(n)}P$$ such that $$\Delta (U)=U$$ for each object $$U$$.
The results of the author are more categorical to state and prove than the shape complement theorems of T. Chapman [Fundam. Math. 76, 181–193 (1972; Zbl 0262.55016) and ibid. 76, 261–276 (1972; Zbl 0222.55019)]. Other attempts as generalization also have had a more geometrical flavor [see R. B. Sher, Lect. Notes Math. 870, 150–168 (1981; Zbl 0494.57007) and ibid. 1283, 212–220 (1987; Zbl 0631.55006)].

##### MSC:
 55P55 Shape theory
Full Text:
##### References:
 [1] F.W. Bauer , Some relations between shape constructions , Cahiers Topologie Géom. Diff. 19 (1978), 337-367. · Zbl 0404.54028 · numdam:CTGDC_1978__19_4_337_0 · eudml:91206 [2] A. Calder and H.M. Hastings , Realizing strong shape equivalences , J. Pure Appl. Algebra 20 (1981), 129-156. · Zbl 0457.55004 · doi:10.1016/0022-4049(81)90088-8 [3] F.W. Cathey , Strong shape theory in: Shape theory and Geometric Topology , (ed. S. Mardešiċ, J. Segal), 215-238 Lecture Notes in Math. 870, Springer, Berlin -Heidelberg-New York 1981. · Zbl 0473.55011 [4] J. Dydak and J. Segal , Strong shape theory , Diss. Math. 192 (1981), 1-42. · Zbl 0474.55007 [5] D.A. Edwards and H.M. Hastings , Čech and Steenrod homotopy theories with applications to geometric topology , Lecture Notes in Math. 542, Springer, Berlin -Heidelberg-New York 1976. · Zbl 0334.55001 · doi:10.1007/BFb0081083 [6] Q. Haxhibeqiri and S. Nowak , Stable shape , Lecture given at the Conference on Geometric Topology and Shape Theory, Dubrovnik 1986. · Zbl 0682.55007 [7] Y. Kodama and J. Ono , On fine shape theory , Fund. Math. 105 (1979), 29-39. · Zbl 0425.54016 · eudml:211079 [8] Yu. T. Lisica , On the exactness of the spectral homotopy group sequence in shape theory , Soviet Math. Dok. 18 (1977), 1186-1190. · Zbl 0398.55012 [9] Yu. T. Lisica and S. Mardešić , Coherent prohomotopy theory and strong shape of metric compacta , Glasnik Mat. 20 (40) (1985), 159-186. · Zbl 0592.55007 [10] S Mardešić and J. Segal , Shape Theory , North-Holland, Amsterdam 1982. · Zbl 0495.55001 [11] P. Mrozik , Hereditary shape equivalences and complement theorems , Topology Appl. 22 (1986), 131-137. · Zbl 0598.54006 · doi:10.1016/0166-8641(86)90003-9 [12] P. Mrozik , Chapman’s category isomorphism for arbitrary ARs , Fund. Math. 125 (1985), 195-208. · Zbl 0592.57013 · eudml:211579 [13] P. Mrozik , Finite-dimensional categorical complement theorems in shape theory , Compositio Math. 68 (1988), 161-173. · Zbl 0665.55006 · numdam:CM_1988__68_2_161_0 · eudml:89933 [14] S. Nowak , On the relationship between shape properties of subcompacta of Sn and homotopy properties of their complements , Fund. Math. 128 (1987), 47- 59. · Zbl 0633.55009 · eudml:211648 [15] D. Quillen , Homotopical Algebra , Lecture Notes in Math. 43, Springer, Berlin -Heidelberg-New York 1967. · Zbl 0168.20903 · doi:10.1007/BFb0097438 [16] R.B. Sher , Complement theorems in shape theory , in: Shape Theory and Geometric Topology (ed. S. Mardešić, J. Segal), 150-168, Lecture Notes in Math. 870, Springer, Berlin-Heidelberg-New York 1981. · Zbl 0494.57007 [17] R.B. Sher , Complement theorems in shape theory II , in: Geometric Topology and Shape Theory (ed. S. Mardešić, J. Segal), 212-220, Lecture Notes in Math. 1283, Springer, Berlin-Heidelberg -New York 1987. · Zbl 0631.55006 [18] E.H. Spanier , Algebraic Topology , McGraw-Hill, New York 1966. · Zbl 0145.43303 [19] R.M. Switzer , Algebraic Topology-Homotopy and Homology , Springer, Berlin 1975. · Zbl 1003.55002 [20] G.A. Venema , Embeddings of compacta in the trivial range , Proc. Amer. Math. Soc. 55 (1976), 443-448. · Zbl 0332.57005 · doi:10.2307/2041743
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