A decomposition theorem for 𝑛-dimensional manifolds

Abstract

Considered from one point of view Theorem 1 is a generalization of Corollary 1 in [3]. From still another the result says that any «-manifold is almost triangulable. The proof of Theorem 1 leads to more interesting results in the case of compact manifolds which we shall consider presently. The steps in the proof will be described here. If Cn is a closed «-cell in M such that Bd Cn, the boundary of Cn, is bicollared in Afn, [2], and if {a,-} is a countable dense subset of Mn\Cn, consider the set CUsi. Does this set lie on the interior of an «-cell in M with a bicollared boundary? If this were the case and if Ci is such an «-cell, one could ask if CiVJo2 lies interior to an «-cell in Mn with a bicollared boundary. Continuing in this way with sets of the form d^Ja^i, if such enclosure is always possible, we obtain an increasing sequence {d) of closed «-cells in Mn, Bd d is bicollared in M and int C?+i Z)Ci, where interior Ci+i is written int d+\. Next we observe that P = U,Ci is En by either a direct construction of cells with annuli between them or by applying the main result of [l]. Then Mn — Pn = C is nowhere dense in Mn and closed since Pn is open. The sets Pn and C would then meet the requirements of Theorem 1. From this outline it is clear that the proof of Theorem 1 follows immediately from a lemma.