CUTTING AND PASTING LIBERT E, EGALIT E, HOMOLOGIE! 1

Transcription

1 CUTTING AND PASTING LIBERTÉ, EGALITÉ, HOMOLOGIE! 1

2 1. Heegaard splittigs Makig 3-maifolds from solid hadlebodies 2. Surface homeomorphisms Gluig maifolds together alog their boudaries 3. Surgery Cuttig ad pastig 4. Homology spheres If it looks like a sphere... it might ot be. 2

3 1. HEEGAARD SPLITTINGS -maifold: Compact, coected, Hausdorff topological space, each poit of which has a eighbourhood homeomorphic to.... with boudary: Also allow eighbourhoods homeomorphic to. boudary, : The bit cosistig of poits with eighbourhoods homeomorphic to. Geus- hadlebody: Compact subset of bouded by a geus- surface (a 2-sphere with hollow hadles). A geus-3 hadlebody 3

4 Now: 1. Take two idetical copies of the geushadlebody. 2. Choose a homeomorphism. 3. Form the quotiet space : Take the disjoit uio ad idetify with its image. This is a Heegaard splittig (of geus 3-maifold. ) of the THEOREM A 3-maifold formed i this way is orietable. Furthermore, ay orietable 3-maifold ca be preseted thusly. 4

5 * % ( EXAMPLES! 1. Geus-0: The 3-sphere. Two copies of glued together alog their boudaries ( ). "! Aalogous to ad. The North ad South poles of are the cetres of the s 2. Geus-1: The 3-sphere (agai). Ca also split as two solid tori ( # ): $)( ' % $&% ' ( * 5

6 5 7 # 3. Glue +. to + ad, to,. 4. Projective space Glue each meridia +0/ to a 21 torus kot o the surface of the other solid torus. 5. Les spaces 50687:9. >= Glue each meridia + ad + to a <; torus kot o the surface of the other torus. This is oe costructio of the les space 5 687:9. I particular, 7?9, # 5A@ 7, ad 5.-. Also, 5 687: :9)B iff C ;EDGF ;IH (mod = ), ad 5J687:9 K 5 687:9)B iff ; F L ;ED (mod = ). For example: 5AM 7 ad 5"M 7 have the same homotopy type, but are ot homeomorphic. 6

7 HOMEOMORPHISMS OF SURFACES Heegaard splittigs of a give geus are determied by the gluig homeomorphism. Wat a descriptio of such i terms of suitable elemetary operatios. DEHN TWISTS Cut the surface aroud a meridioal curve, twist, ad glue back together agai. THEOREM (DEHN-LICKORISH) Ay orietatio-preservig homeomorphism of a orieted 2-maifold (without boudary) is isotopic to a compositio of Deh twists. 7

8 O COROLLARY Ay orietable 3-maifold ca be costructed by cuttig out a collectio of ukotted solid tori from ad gluig them back i alog differet boudary homeomorphisms. COROLLARY (ROKHLIN S THEOREM) Every orietable 3-maifold (without boudary) is the boudary of a 4-maifold. That is, N. 8

9 @ S RATIONAL SURGERY Take, cut out a ukotted solid torus with meridia + ad logitude,. The glue it back i by idetifyig + with the curve = + P ;E,, where = ad ; are Q= coprime (this is a <; torus kot). This surgery is determied completely by the ratioal umber R 9, which we call the framig idex 6 of the ukotted torus. EXAMPLES 1. S #. This is a torus switch 2. S TU 5 687:9. 3. S H WV. 4. S X Y W[Z V \ V 9

10 ^ O O LINKING NUMBERS To geeralise to otrivial kots, we eed to be a bit more careful whe choosig the logitude. Give two curves ] ad ^ i, defie their likig umber _a`bc] ^ to be the sum, over all the crossigs d, of edf : J J K ε = +1 K ε = 1 Crossigs which do t ivolve two differet compoets of the lik have e. _g`bc] _g`g4^ ]. If _g`gc] ^ h the ] ad ^ are liked. The coverse is t true i geeral, though. The likig umber is ivariat uder the Reidemeister moves: It s idepedet of the isotopy class of the lik. 10

11 O Now, give a kot ], choose a meridia + o its tubular eighbourhood such that _aì<+ ]J P 1, ad a logitude, which is codirected with ] such that _g`bj, ]. J β α INTEGER SURGERY We ca ow do ratioal surgery o otrivial liks. It turs out, though, that iteger surgery is eough: THEOREM Ay compact, orietable 3-maifold without boudary ca be obtaied by iteger surgery o a lik i. 11

12 EQUIVALENT SURGERIES Surgery o alog differet framed liks ca produce homeomorphic maifolds. Two such surgeries are said to be equivalet. THE KIRBY CALCULUS A k -move cosists of addig or deletig a uliked trivial kot with framig C 1 : -+1 A k -move cosists of a hadle-slide: k k+ 12

13 THEOREM (KIRBY) Two liks i with iteger framigs produce the same 3-maifold iff they ca be obtaied from each other by a fiite sequece of Kirby moves ad isotopies. FENN-ROURKE MOVES A Fe-Rourke move is as follows: +1 (If the circle has framig l other way.) 1, the the kiks go the THEOREM (FENN-ROURKE) A framed lik m ca be trasformed by Kirby moves ito the framed lik m iff this ca be doe by Fe- Rourke moves. 13

14 s q q O THE FUNDAMENTAL GROUP The fudametal group of a topological space o, deoted o is essetially a way of coutig the (1-dimesioal) holes i o. Its elemets are the homotopy classes of based loops (maps o ) i o, with the multiplicatio operatio beig give by cocateatio ad the idetity beig the loop which ca be shruk dow to the basepoit. For example:, because every loop ca be shruk dow # to the basepoit. But p, homotopy classes of loops beig determied by the umber of times they wid aroud the cetral hole. Ad r5 687:9 6. HOMOLOGY GROUPS s The homology groups s o are aother way of coutig the -dimesioal holes i o. I particular, part of Hurewicz theorem says that, if o is path-coected: PROPOSITION o o utgvw o o yx. That is, the first homology of o is the same as the abeliaisatio of the fudametal group. 14

15 s s s s s q O HOMOLOGY 3-SPHERES A homology 3-sphere is a compact, pathcoected 3-maifold (without boudary), which has the same series of homology groups as Or (by Poicaré duality ad the UCT): is trivial. Or (by the above fragmet of Hurewicz theorem): coicides with its commutator subgroup {} ~& i ~ z4~ vw yx Eƒ CONJECTURE (POINCARÉ) Every homology 3-sphere is homeomorphic to. 15

16 @ This is false, leadig Poicaré to suggest: CONJECTURE (POINCARÉ) Every homotopy 3-sphere is homeomorphic to. POINCARÉ S HOMOLOGY 3-SPHERE This is a 3-maifold with trivial s but otrivial (ad is hece ot homeomorphic to ). May differet costructios... (cf. Kirby ad Scharlema: Eight faces of the Poicaré homology 3-sphere) DODECAHEDRAL SPACE Take a solid dodecahedro ad idetify opposite faces with a twist. 16

17 ^ SURGERY ON THE TREFOIL with fram- Do surgery o ig +1. alog the right trefoil ^ +1 This gives a maifold which is t homeomorphic to, because 4 is otrivial. CALCULATION OF 4 First, calculate the fudametal group of the complemet. This is geerated by three loops rˆ <Š z * x y 17

18 Œ ˆ ˆ ˆ Š ˆ ˆ ˆ ^ ˆ ˆ subject to the relatios ˆ ˆ Š Š z : x x y givig a presetatio 4ˆ ˆŽ Whe we glue i the solid torus, we attach its meridioal disk to the logitude of the tubular eighbourhood of the (removed) trefoil, with framig +1. obtaied corre- Thus, 4 is the quotiet of by killig the word spodig to this logitude: Œ rˆ 4 p ˆ rˆ 18

19 h ˆ By substitutig 4~ form: Œ z4~ ~, we get the eater ~& This group (the biary icosahedral group) is otrivial (it has order 120 ad is the isometry group of the icosahedro), hece, but it has trivial abeliaisatio. SURGERY ON THE WHITEHEAD LINK 1 1 SURGERY ON THE BORROMEAN RINGS