Reference request for wild 3-manifolds












4















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?






reference-request gt.geometric-topology 3-manifolds







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    4















    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?






    reference-request gt.geometric-topology 3-manifolds







    share|cite|improve this question



























      4












      4








      4


      4






      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?






      reference-request gt.geometric-topology 3-manifolds







      share|cite|improve this question
















      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?






      reference-request gt.geometric-topology 3-manifolds




      reference-request gt.geometric-topology 3-manifolds






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      edited 5 hours ago









      Martin Sleziak

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      asked 5 hours ago









      James BaxterJames Baxter

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          6














          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".






          share|cite|improve this answer
























          • Thanks very much, I’ll check out all those three you mentioned at the end!

            – James Baxter
            5 hours ago



















          2














          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.






          share|cite|improve this answer































            0














            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.






            share|cite|improve this answer























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              3 Answers
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              3 Answers
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              active

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              active

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              active

              oldest

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              6














              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".






              share|cite|improve this answer
























              • Thanks very much, I’ll check out all those three you mentioned at the end!

                – James Baxter
                5 hours ago
















              6














              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".






              share|cite|improve this answer
























              • Thanks very much, I’ll check out all those three you mentioned at the end!

                – James Baxter
                5 hours ago














              6












              6








              6







              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".






              share|cite|improve this answer













              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".







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 5 hours ago









              Andy PutmanAndy Putman

              31.4k6133213




              31.4k6133213













              • Thanks very much, I’ll check out all those three you mentioned at the end!

                – James Baxter
                5 hours ago



















              • Thanks very much, I’ll check out all those three you mentioned at the end!

                – James Baxter
                5 hours ago

















              Thanks very much, I’ll check out all those three you mentioned at the end!

              – James Baxter
              5 hours ago





              Thanks very much, I’ll check out all those three you mentioned at the end!

              – James Baxter
              5 hours ago











              2














              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.






              share|cite|improve this answer




























                2














                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.






                share|cite|improve this answer


























                  2












                  2








                  2







                  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.






                  share|cite|improve this answer













                  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.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  Ian AgolIan Agol

                  48.9k3126235




                  48.9k3126235























                      0














                      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.






                      share|cite|improve this answer




























                        0














                        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.






                        share|cite|improve this answer


























                          0












                          0








                          0







                          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.






                          share|cite|improve this answer













                          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.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 2 hours ago









                          Scott TaylorScott Taylor

                          32925




                          32925






























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