I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read through Schulten’s Introduction to 3 manifolds. Are there any good texts about these that focus on the geometric aspects more than the algebraic side?
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3 Answers
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The only two books that I know of that focus on wild/pathological aspects of 3-manifolds are Bing's "Geometric topology of 3-manifolds" and Moise's "Geometric topology in dimension 2 and 3". But if you want to learn "Bing style topology", the best sources expand their focus to high-dimensional phenomena as well. Here three rather different (but complementary) textbooks are Daverman's "Decompositions of Manifolds", Rushing's "Topological embeddings", and Daverman-Venema "Embeddings in Manifolds".
answered Jan 11, 2019 at 16:32
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$\begingroup$ Thanks very much, I’ll check out all those three you mentioned at the end! $\endgroup$ Commented Jan 11, 2019 at 16:44
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As far as I know, there's not really any textbooks on this subject. There's a memoir by Brin and Thickstun, but this is not focussed on geometric aspects of wild manifolds.
Sylvain Maillot has explored non-compact 3-manifolds, and in particular studied Ricci flow on manifolds of bounded geometry.
Tommaso Cremaschi has a couple of papers on non-compact 3-manifolds which do not admit a hyperbolic metric.
In a paper of John Pardon on the Hilbert-Smith conjecture, he analyzes certain wild manifolds with two ends, and shows that the (istopy classes of) closed incompressible surfaces which separate the ends form a lattice. The proof of this uses some geometry of minimal surfaces.
The literature on wild 3-manifolds is sparse enough that no one has taken up the effort to write a textbook on the subject as far as I know. There is no conjectural classification; indeed it is hard to imagine what such a classification might look like.
answered Jan 11, 2019 at 19:52
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One interpretation of wild/pathological in the 3-manifold setting is noncompact 3-manifolds. Peter Scott and Thomas Tucker have a few lovely papers and in the decades since there have been papers appearing every so often. Myers also has a series of papers on the ends of noncompact 3-manifolds. I'll take the opportunity to advertise a paper on non-compact Heegaard splittings which studies "wild" and not-so wild behavior of non-compact Heegaard splittings.
My impression is that the geometry of such things has been mostly focused on showing that under certain algebraic or geometric hypotheses things (such as ends) that could be wild actually aren't. I'm thinking of the work of Agol and Calegari-Gabai, but there's a lot of other important work too.
