By S. Zaidman

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Homeomorphic to S4, "The general case can be reduced to the case IK21 CM by addition of a collar to OM; compare footnote 32. 36Compare the preceding footnote. 44 Hog-Angeloni/Metzler: I. l. homeomorphic to 2N4. l. homeomorphic to S4. 4 for M4 x I alternatively could be deduced from the collapses M4 x I \, H x I \ *; compare (56) above. 36). 1. 1. 4-manifold. As opposed to the 3dimensional situation, the multitude (not the existence) of regular manifold neighbourhoods is the central topic in dimension 4.

A systematic treatment of this phenomenon is a main topic of Chapter III. Presentation classes even contain genuine simple-homotopy information, as simple-homotopy type and homotopy type differ already in dimension 2, see the references given in (21) above. At the time of writing, only (30) contributes a question mark in the following chain of implications: (41) K2 L2 ? K2 / L2 K2 L2 ir1(IK21) zz ir1(IL21) and X(IK21) = X(IL21), where K2, L2 are finite, connected CW-complexes. Because of (40), all distinguishing examples necessarily have a "small" Euler characteristic; in many cases the minimal value of X is required; see Chapters III, VII and IX.

This implies that again r embeds in S2 in a way that permits the thickening of K2 to a 3-manifold. Remarks: 1) By similar techniques, with the use of an additional permutation B, Neuwirth's algorithm is able to detect whether there is a thickening of K2 whose boundary is a 2-sphere, thus answering (48) for being a spine of a closed orientable 3-manifold; see [Ne68]. 2) The interested reader may extend the algorithm to the nonorientable case. We now turn our attention to question (49), to which the answer is also "no".