4 dimensional analogues of Dehn s lemma
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1 4 dimensional analogues o Dehn s lemma Arunima Ray Brandeis University Joint work with Daniel Ruberman (Brandeis University) Joint Mathematics Meetings, Atlanta, GA January 5, 2017 Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
2 Classical Dehn s lemma in three dimensions Theorem (Dehn s lemma) Any nullhomotopic embedded circle in the boundary o a 3 maniold extends to a map o an embedded disk. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
3 Classical Dehn s lemma in three dimensions Theorem (Dehn s lemma) Any nullhomotopic embedded circle in the boundary o a 3 maniold extends to a map o an embedded disk. i.e. given Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
4 Classical Dehn s lemma in three dimensions Theorem (Dehn s lemma) Any nullhomotopic embedded circle in the boundary o a 3 maniold extends to a map o an embedded disk. i.e. given Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
5 Classical Dehn s lemma in three dimensions Theorem (Dehn s lemma) Any nullhomotopic embedded circle in the boundary o a 3 maniold extends to a map o an embedded disk. i.e. given S 1 M 3 D 2 M 3 embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
6 Classical Dehn s lemma in three dimensions Theorem (Dehn s lemma) Any nullhomotopic embedded circle in the boundary o a 3 maniold extends to a map o an embedded disk. i.e. given S 1 M 3 D 2 M 3 embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
7 Classical Dehn s lemma in three dimensions S 1 M 3 D 2 M 3 embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
8 Classical Dehn s lemma in three dimensions S 1 M : stated by Dehn D 2 M 3 embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
9 Classical Dehn s lemma in three dimensions S 1 M 3 D 2 M 3 embedding 1910: stated by Dehn 1929: error ound in Dehn s proo by Kneser Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
10 Classical Dehn s lemma in three dimensions S 1 M 3 D 2 M 3 embedding 1910: stated by Dehn 1929: error ound in Dehn s proo by Kneser 1957: correct proo given by Papakyriakopoulos Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
11 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
12 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 1: Consider embedded circles in the boundary o 4 maniolds. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
13 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 1: Consider embedded circles in the boundary o 4 maniolds. That is, i an embedded circle in the boundary o a 4 maniold is nullhomotopic in the interior, does it bound an embedded disk?? Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
14 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 1: Consider embedded circles in the boundary o 4 maniolds. That is, i an embedded circle in the boundary o a 4 maniold is nullhomotopic in the interior, does it bound an embedded disk?? This is a question about slice knots, which are widely studied. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
15 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
16 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres. S 2 W 4 D 3 W 4? embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
17 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres. S 2 W 4 D 3 W 4? embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
18 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres or tori. S 2 W 4 S 1 S 1 W 4 D 3 W 4 W 4? embedding? embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
19 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres or tori. S 2 W 4 S 1 S 1 W 4 D 3 W 4? embedding S 1 D 2 W 4? embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
20 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres or tori. S 2 W 4 S 1 S 1 W 4 D 3 W 4? embedding S 1 D 2 W 4? embedding Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
21 Goal Question Is there an analogue o Dehn s lemma in our dimensions? Possibility 2: Consider codimension one submaniolds o the boundary o 4 maniolds, e.g. spheres or tori. S 2 W 4 S 1 S 1 W 4 D 3 W 4? embedding S 1 D 2 W 4? embedding Moreover, we can ask whether these embeddings exist smoothly or merely topologically (i.e. locally lat). Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
22 Results Theorem (R. Ruberman) or embedded spheres/tori in the boundary o 4 maniolds, Dehn s lemma 1 does not hold in general Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
23 Results Theorem (R. Ruberman) or embedded spheres/tori in the boundary o 4 maniolds, Dehn s lemma 1 does not hold in general 2 holds under certain broad hypotheses Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
24 Results Theorem (R. Ruberman) or embedded spheres/tori in the boundary o 4 maniolds, Dehn s lemma 1 does not hold in general 2 holds under certain broad hypotheses 3 sometimes holds topologically but not smoothly Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
25 Results or spheres Theorem (R. Ruberman) There exists a sphere S W 4 where W is smooth and simply connected and S is nullhomotopic in W, but S does not bound a topological ball in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
26 Results or spheres Theorem (R. Ruberman) There exists a sphere S W 4 where W is smooth and simply connected and S is nullhomotopic in W, but S does not bound a topological ball in W. Theorem (R. Ruberman) I Y = Y 1 # S Y 2 = W 4 where Y 2 is an integer homology sphere, π 1 (W ) is good, and π 1 (Y 2 ) π 1 (W ) is the trivial map, then S bounds a topologically embedded ball in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
27 Results or spheres Theorem (R. Ruberman) There exists a sphere S W 4 where W is smooth and simply connected and S is nullhomotopic in W, but S does not bound a topological ball in W. Theorem (R. Ruberman) I Y = Y 1 # S Y 2 = W 4 where Y 2 is an integer homology sphere, π 1 (W ) is good, and π 1 (Y 2 ) π 1 (W ) is the trivial map, then S bounds a topologically embedded ball in W. Corollary (R. Ruberman) Any sphere S Y = W 4 where Y is an integer homology sphere and π 1 (W ) is abelian bounds a topologically embedded ball in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
28 Results or spheres Corollary (R. Ruberman) Any sphere S Y = W 4 where Y is an integer homology sphere and π 1 (W ) is abelian bounds a topologically embedded ball in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
29 Results or spheres Corollary (R. Ruberman) Any sphere S Y = W 4 where Y is an integer homology sphere and π 1 (W ) is abelian bounds a topologically embedded ball in W. Theorem (R. Ruberman) There exists a sphere S Y = W 4 with W smooth and simply connected and Y an integer homology sphere such that S bounds a topologically embedded ball in W but no smooth ball in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
30 Results or spheres Corollary (R. Ruberman) Any sphere S Y = W 4 where Y is an integer homology sphere and π 1 (W ) is abelian bounds a topologically embedded ball in W. Theorem (R. Ruberman) There exists a sphere S Y = W 4 with W smooth and simply connected and Y an integer homology sphere such that S bounds a topologically embedded ball in W but no smooth ball in W. Example: Let P be the Poincaré homology sphere with a disk removed and γ a curve that normally generates π 1 (P ). Let W be the 4 maniold obtained rom P [0, 1] by doing surgery along γ pushed into the interior. Then W = P #P, where the connected-sum is perormed along a sphere S. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
31 Results or tori Theorem (R. Ruberman) There exists an incompressible torus T Y = W 4 where W is contractible such that T extends to a map o the solid torus to W, but does not bound an embedded solid torus in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
32 Results or tori Theorem (R. Ruberman) There exists an incompressible torus T Y = W 4 where W is contractible such that T extends to a map o the solid torus to W, but does not bound an embedded solid torus in W. Proposition (R. Ruberman) Let T Y = W be a separating torus, γ T a simple closed curve, and e the surace induced raming. I 1 γ is non-trivial in H 1 (T ), 2 γ is smoothly (resp. topologically) slice in W with respect to e, and 3 the surgered maniold Y e (γ) is irreducible, then T bounds a smoothly (resp. topologically) embedded solid torus in W. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
33 Results or tori Theorem (R. Ruberman) There exists a contractible W and an incompressible torus T Y = W such that T extends to a topological embedding o a solid torus in W, but not a smooth embedding. T 0 K J J Here J is the right-handed treoil and K is the positive untwisted Whitehead double o the right-handed treoil. Arunima Ray (Brandeis) 4 dimensional analogues o Dehn s lemma January 5, / 9
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