## Detecting codimension one manifold factors with the disjoint homotopies property.(English)Zbl 0992.57024

A generalized $$n$$-manifold $$X$$ is a finite-dimensional, locally contractible metric space such that $$H_n(X,X \smallsetminus \{x\};\mathbb{Z}) \cong \mathbb{Z}$$ and otherwise $$H_i(X,X \smallsetminus\{x\}; \mathbb{Z})\cong 0$$; such a space is said to be resolvable if there exist an $$n$$-manifold $$M$$ and a cell-like mapping $$f$$ of $$M$$ onto $$X$$ $$(f$$ is cell-like provided for each $$x\in X$$ and neighborhood $$U$$ of $$f^{-1}(x)$$, the inclusion $$f^{-1}(x) \hookrightarrow U$$ is null-homotopic). An approximately 40 year old problem traceable to R. H. Bing asks whether, for every resolvable generalized $$n$$-manifold $$X$$, $$X\times \mathbb{R}$$ is a manifold. If so, then $$X$$ is said to be a codimension one manifold factor. It is known that in this setting $$X\times\mathbb{R}^2$$ always is an $$(n+2)$$-manifold. The reviewer proved [Pac. J. Math. 93, 277-298 (1981; Zbl 0415.57007)] that a generalized $$n$$-manifold $$X$$ $$(n>3)$$ is a codimension one manifold factor if each pair of maps $$f:I\to X$$, $$g:B^2\to X$$ can be approximated by maps $$F:I\to X$$, $$G:B^2\to X$$ with disjoint images. In what may be the first result on the topic since then, here the author introduces a Disjoint Homotopies Property and shows that all generalized $$n$$-manifolds $$X$$ $$(n>3)$$ having it are codimension one manifold factors; $$X$$ has the Disjoint Homotopies Property if any two homotopies $$f_t,g_t: I\to X$$ can be approximated by homotopies $$F_t,G_t:I\to X$$ such that $$F_t(I)\cap G_t(I) =\emptyset$$ for all $$t$$.
The author introduces another property, the Plentiful 2-manifolds Property, which (for metric spaces $$X)$$ is characterized by the fact that each path in $$X$$ can be approximated by another path which lies in a 2-manifold $$N \subset X$$. Then in another key result the author establishes that all resolvable generalized $$n$$-manifolds having the Plentiful 2-manifolds Property are codimension one manifold factors. She concludes by constructing a new class of codimension one manifold factors, namely, the $$k$$-ghastly examples $$(2<k<n)$$, which contain no embedded $$k$$-cells but do contain embedded $$(k-1)$$-cells; these are constructed so as to have the Plentiful 2-manifolds Property.

### MSC:

 57P05 Local properties of generalized manifolds 54B10 Product spaces in general topology 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010)

Zbl 0415.57007
Full Text:

### References:

 [1] Antoine, L., Sur l’homéomorphie de deux figures et de leurs voisinages, J. math. pures appl. (9), 86, 221-325, (1921) · JFM 48.0650.01 [2] F.D. Ancel, The locally flat approximation of cell-like embedding relations, Thesis, 1976-1977 [3] Bing, R.H., Upper semicontinuous decompositions of E3, Ann. of math. (2), 65, 363-374, (1957) · Zbl 0078.15201 [4] Bing, R.H., A decomposition of E3 into points and tame arcs such that the decomposition space is topologically different from E3, Ann. of math. (2), 65, 484-500, (1957) · Zbl 0079.38806 [5] Bing, R.H., The Cartesian product of a certain non-manifold and a line is E4, Ann. of math. (2), 70, 399-412, (1959) · Zbl 0089.39501 [6] Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S., Topology of homology manifolds, Ann. of math. (2), 143, 435-467, (1996) · Zbl 0867.57016 [7] Cannon, J.W., The recognition problem: what is a topological manifold?, Bull. amer. math. soc., 84, 832-866, (1978) · Zbl 0418.57005 [8] Cannon, J.W., Shrinking cell-like decomposition of manifolds. codimension three., Ann. of math. (2), 110, 83-112, (1979) · Zbl 0424.57007 [9] Cannon, J.W.; Bryant, J.L.; Lacher, R.C., The structure of generalized manifold having nonmanifold set of trivial dimension, (), 261-300 · Zbl 0476.57006 [10] Cannon, J.W.; Daverman, R.J., A totally wild flow, Indiana univ. math. J., 30, 371-387, (1981) · Zbl 0432.58018 [11] Cohen, M.M., Simplicial structures and transverse cellularity, Ann. of math. (2), 85, 218-245, (1967) · Zbl 0147.42602 [12] Daverman, R.J., Products of cell-like decompositions, Topology appl., 11, 121-139, (1980) · Zbl 0436.57005 [13] Daverman, R.J., Detecting the disjoint disks property, Pacific J. math., 93, 277-298, (1981) · Zbl 0415.57007 [14] Daverman, R.J., Decompositions of manifolds, (1986), Academic Press New York · Zbl 0608.57002 [15] Daverman, R.J.; Edwards, R.D., Wild Cantor sets as approximations to codimension two manifolds, Topology appl., 26, 207-218, (1987) · Zbl 0624.57021 [16] Daverman, R.J.; Walsh, J.J., A ghastly generalized n-manifold, Illinois J. math., 25, 555-576, (1981) · Zbl 0478.57014 [17] Eaton, W.T., A generalization of the dog bone space to En, Proc. amer. math. soc., 39, 379-387, (1973) · Zbl 0262.57001 [18] Freedman, M.H., The topology of four-dimensional manifolds, J. differential geom., 17, 357-453, (1982) · Zbl 0528.57011 [19] D.M. Halverson, Detecting codimension one manifold factors with the disjoint homotopies property, Thesis, 1999 · Zbl 0992.57024 [20] Haver, W.E., Mappings between ANRs that are fine homotopy equivalences, Pacific J. math., 58, 457-461, (1975) · Zbl 0311.55006 [21] Lacher, R.C., Cell-like mappings and their generalizations, Bull. amer. math. soc., 83, 495-552, (1977) · Zbl 0364.54009 [22] (), 21-26, McAuley, Upper semicontinuous decompositions of E3 into E3 and generalizations to metric spaces [23] McMillan, D.R., A criterion for cellularity in a manifold, Ann. of math. (2), 79, 327-337, (1964) · Zbl 0117.17102 [24] Moore, R.L., Concerning upper semicontinuous collections of continua which do not separate a given set, Proc. nat. acad. sci., 10, 356-360, (1924) [25] Moore, R.L., Concerning upper semicontinuous collections of compacta, Trans. amer. math. soc., 27, 416-428, (1925) · JFM 51.0464.03 [26] Quinn, F., An obstruction to the resolution of homology manifolds, Michigan math. J., 34, 285-291, (1987) · Zbl 0652.57011 [27] Rourke, C.P.; Sanderson, B.J., Introduction to piecewise-linear topology, Ergebn. math. grenzgeb. (3), 69, (1972), Springer-Verlag Berlin · Zbl 0254.57010 [28] Stone, A.H., Metrizability of decomposition spaces, Proc. amer. math. soc., 7, 690-700, (1956) · Zbl 0071.16001 [29] Wilder, R., Topology of manifolds, Amer. math. soc. colloq. publ., 32, (1963), American Mathematical Society Providence, RI · Zbl 0117.16204 [30] Whyburn, G.T., On the structure of continua, Bull. amer. math soc., 42, 49-73, (1936) · JFM 62.0691.05
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.