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- W2006402022 abstract "IN HIS remarkable paper [ 121 A. N. DraniSnikov gave a method for constructing for every n 2 3 examples of infinite-dimensional compacta (i.e. compact metric spaces) X, with integral cohomological dimension c-dimzX, = n. It follows by a well-known result of R. D. Edwards [34] that there are therefore n-dimensional compacta Y, and cell-like surjections f.: Y. -+ X,, i.e. for every XE X,, the pre-image f;‘(x) has trivial shape. By the NiibelungPontrjagin embedding theorem [29] there exist embeddings 4.: Y, + IW2”+r which in turn yield upper semicontinuous decompositions G, of R2”+ ‘, with non-degeneracy set given by {4,/i *(x)IxeX,}, whose quotient spaces R 2n+‘/G contain X, and so are infinite dimenn sional. Thus cell-like maps can raise dimension on manifolds of dimensions 7 and above. Such phenomena arc impossible in Iw4 for q z$ 3. Cell-like images of topological qmanifolds are always Z-homology q-manifolds [36], and for these cohomoiogical and covering dimensions agree if q I; 3; for q 5 2 it is classical that homology q-manifolds are topological manifolds [37], for q = 3 see [353. Recently J. Dydak and J. J. Walsh [16] have shown that there exists an infinite-dimensional compactum X with c-dimzX = 2. The preceding argument shows that cell-like maps can also raise dimension on Iws and R6. For more on the cell-like mapping problem and its history, see the survey [25]. We are interested in dimension four, the remaining unsettled case of the cell-like mapping problem. A. N. DraniSnikov and E. V. Scepin conjectured Cl53 that cell-like maps cannot raise dimension on 4-manifolds. As evidence for this conjecture we prove:" @default.
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- W2006402022 title "On 1-cycles and the finite dimensionality of homology 4-manifolds" @default.
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- W2006402022 doi "https://doi.org/10.1016/0040-9383(92)90054-l" @default.
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