aa r X i v : . [ m a t h . G T ] J a n THE AKHMEDOV-PARK EXOTIC CP CP SELMAN AKBULUT
Abstract.
Here we draw a handlebody picture for the exotic CP CP constructed by Akhmedov and Park. Introduction
Akhmedov-Park manifold M is a symplectic 4-manifold which is an exoticcopy of CP CP ([AP], also see related [Ak]). It is obtained from twocodimension zero pieces which are glued along their common bondaries: M = ˜ E ⌣ ∂ ˜ E The two pieces are constructed as follows: We start with the product ofa genus 2 surface and the torus E = Σ × T . Let < a , b , a , b > and < C, D > be the standard circles generating the first homology of Σ and T respectively (the cores of the 1-handles). Then ˜ E is obtained from E by doing “Luttinger surgeries” to the four subtori ( a × C, a ), ( b × C, b ),( a × C, C ) ( a × D, D ) (see Section 1 for Luttinger surgery). Then˜ E = ˜ E − Σ × D To built the other piece, let K ⊂ S be the trefoil knot, and S ( K ) be the3-manifold obtained doing 0-surgery to S along K . Being a fibered knot, K induces a fibration T → S ( K ) → S and the fibration T → S ( K ) × S → T Let T and T be the vertical (fiber) and the horizontal (section) tori ofthis fibration, intersecting at one point p . Smoothing T ∪ T at p givesan imbedded genus 2 surface with self intersection 2, hence by blowing upthe total space twice (at points on this surface) we get a genus 2 surfaceΣ ⊂ ( S ( K ) × S ) CP with trivial normal bundle. Then we define:˜ E = ( S ( K ) × S ) CP − Σ × D In short M = ˜ E ♯ Σ ( S ( K ) × S ) CP ( ♯ Σ denotes fiber sum along Σ ).Alternatively, M can be built by using Sym (Σ ) [FPS], which is equivalentto this construction, since Sym (Σ ) = (Σ × T ) CP ♯ Σ ( T CP ) andby Luttinger surgeries this can be transform to M defined above. Also [BK]gives another exotic CP CP which turns out to be a version of this M (we thank Anar Akhmedov for explaining these equivalences). Mathematics Subject Classification.
Remark 1.
Chronologically, first Fintushel-Stern had the idea of buildingexotic CP CP from Sym (Σ ) by Luttinger surgeries, but they couldn’tget their manifold to be simply connected. Then Akhmedov-Park came outwith this exotic CP CP [AP] (subject of this paper). Later in [FPS] Fintushel-Park-Stern fixed the fundamental group problem in their
Sym (Σ ) approach, thereby getting another exotic CP CP and in [BK] Baldridge-Kirk came out with their version. In retrospect, they all are related. Luttinger surgery
Let T ⊂ X be a smooth 4-manifold with an imbedded subtorus whichhas trivial normal bundle ν ( T ) ≈ T × B . Let ϕ p ( p ≥
0) be the self-diffeomorphism of T = T × ∂B given by the matrix in terms of theautomorphism of its standard homology generators ( a, b, c ). p −
10 1 0 The operation of removing ν ( T ) from X and regluing T × B by the map ϕ p : S × T → ∂ν ( T ) is called the p log-transform of X along T . When( X, ω ) is symplectic and T is Lagrangian and p = ±
1, this operation pre-serves symplectic structure and is equivalent to a
Luttinger surgery up todiffeomorphism (e.g. [A3]). We will refer this operation as ( a × b, b ) Luttingersurgery. Figure 1 describes this as a handlebody operation (cf. [AY]) Figure 1.
Luttinger surgery ( p = ± HE AKHMEDOV-PARK EXOTIC CP CP Constructing ˜ E Let E = Σ × T be the surface of genus two crossed by the puncturedtorus. Recall that Figure 2 describes a handlebody picture of E and thebounday identification f : ∂E ≈ −→ Σ × S , as shown in [A1]. The knowl-edge of where the arcs in the figure of E (top of Figure 1) thrown by thediffeomorphism f is essential to our construction. By performing the indi-cated handle slide to E (indicated by dotted arrow) in Figure 3, we obtaina second equivalent picture of E . By performing Luttinger surgeries toFigure 3 along ( a × C, a ), ( b × C, b ) we obtain the first picture of Figure4, and then by handle slides obtain the second picture. First by an isotopythen a handle slide to Figure 4 we obtain the first and second pictures inFigure 5. By a further isotopy we obtain the first picture of Figure 6, andthen by Luttinger surgeries to ( a × C, C ), ( a × D, D ) we obtain the secondpicture in Figure 6. By introducing canceling handle pairs we express thislast picture by a simpler looking first picture of Figure 7. Then by indicatedhandle slides we obtain the second picture of Figure 7, which is ˜ E .2.1. Calculating π ( ˜ E ) . By the indicted handle slide to Figure 7 we ob-tain Figure 8, which is another picture of ˜ E . In this picture we also in-dicated the generators of its fundamental group: { a, b, c, d, e, f, g, h, k, p, q } .We can read off the relations by tracing the attaching knots of the 2-handles(starting at the points indicated by small circles). We get the followingrelations given by the words: af, ab − , dp, cqd − , kh − , bef e − , he − k − e,aqa − d − , ac − a − cp − , ag − a − h − g, gf − kg − k − ,dk − b − pbk, dq − k − b − c − bk, cdc − bf g − egf − e − b − d − After eliminating f, b, p, d, k by using the obvious short words we get: π ( ˜ E ) = * a, c, e, g, h, q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a, e ] = 1 , [ h, e ] = 1 , [ a, q ] = c, [ c − , a ] = cq, [ g, a ] = h, [ g − , h ] = a, [ cq, ah ] = 1 , [ c, ah ] = 1 , [ q − , c ][ g − , e ] = 1 + Constructing ˜ E and M The first picture of Figure 9 is the handlebody of S ( K ) × S ([A2]),where in this picture the horizontal T × D is clearly visible, but not thevertical torus T (which consits of the Seifet surface of K capped off by the2-handle given by the zero framed trefoil). In the second picture of Figure9 we redraw this handlebody so that both vertical and horizontal tori are SELMAN AKBULUT clearly visible (reader can check this by canceling 1- and 2- handle pairs fromthe second picture to obtain the first picture). The first picture of Figure10 gives ( S ( K ) × S ) CP , where by sliding the 2-handle of T over the2-handle of T (and by sliding over the two ¯ CP ’s of the ¯ CP ’s) we obtainedthe imbedding Σ × D ⊂ ( S ( K ) × S ) CP . By isotopies and handleslides (indicated by dotted arrows) we obtain the second picture in Figure10, and then the Figures 11 and 12. Either pictures of Figure 12 representhandlebodies of ( S ( K ) × S ) CP (both have different advantages). Fig-ure 13 is the same as the first picture of Figure 12, drawn in an exaggeratedway so that Σ × D is clearly visible. We now want to remove this Σ × D from inside this handlebody of ( S ( K ) × S ) CP and replace it with ˜ E .The arcs in Figure 2 (describing the diffeomorphism f ) and also in all thesubsequent Figures 3 to 7 show us how to do this, resulting with the handle-body picture of M in Figure 14. Clearly Figure 15 is another handlebodyof M (where we used the second picture of Figure 12 instead of the first).3.1. Checking π ( M ) = 0 . Clearly we can calculate π ( M ) from the pre-sentation of π ( ˜ E ) from Figure 8 (Section 2.1) by introducing new genera-tors x and y (Figure 13 and 14) and 7 new relations coming from the new2-handles in Figure 14. The 7 new relations are given by the words:( xyx ) − ( yxy ) , q − y, g − xy − x − y,e − yx − , cyx − c − xy − , ege − g − , a − h − g − hg After eliminating x and y from the two short relations we get: π ( M ) = * a, c, e, g, h, q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ a, e ] = 1 , [ h, e ] = 1 , [ a, q ] = c, [ c − , a ] = cq, [ g, a ] = h, [ g − , h ] = a, [ cq, ah ] = 1 , [ c, ah ] = 1 , [ q − , c ][ g − , e ] = 1 a = [ h − g − ] , [ e, g ] = 1 , [ c, e ] = 1 g = e − qeq, q = eqe − qe + From this one can easily check π ( M ) = 0. For example, since q = c − [ c − , a ],and a and c commutes with e , then q commutes with e . Hence the last tworelations imply e = 1 and g = q . The relations [ cq, ah ] = 1 and [ c, ah ] = 1imply [ q, ah ] = 1, hence [ g, ah ] = 1, in tern this together with [ g − , h ] = a gives a = 1, then relations [ a, q ] = c and [ g, a ] = h gives c = 1 and h = 1,and hence [ c − , a ] = cq implieq q = 1 and so g = q = 1. HE AKHMEDOV-PARK EXOTIC CP CP Figure 2. E and f : ∂E ≈ −→ Σ × S Figure 3.
Handle slide (the pair of thick arrows indicatewhere we will perform Luttinger surgeries next)
HE AKHMEDOV-PARK EXOTIC CP CP Figure 4.
First we performed Luttinger surgeries along( a × C, a ), ( b × C, b ), then handle slides (thick arrowsindicate where we will perform Luttinger surgeries next) SELMAN AKBULUT
Figure 5.
Isotopy and a handle slides
HE AKHMEDOV-PARK EXOTIC CP CP Figure 6.
Luttinger surgeries along ( a × C, C ), ( a × D, D ) Figure 7.
More isotopies and handle slides and getting ˜ E CP CP Figure 8. ˜ E , with generators of π ( ˜ E ) are indicated Figure 9. S ( K ) × S CP CP Figure 10. ( S ( K ) × S ) CP
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Figure 11. ( S ( K ) × S ) CP CP CP Figure 12. ( S ( K ) × S ) CP
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Figure 13. ( S ( K ) × S ) CP CP CP Figure 14. M Figure 15. M HE AKHMEDOV-PARK EXOTIC CP CP References [A1] S. Akbulut,
Catanese-Ciliberto-Mendes Lopes surface, arXiv:1101.3036.[A2] S. Akbulut,
A fake cusp and a fishtail,
Turkish Jour. of Math 1 (1999), 19-31.arXiv:math.GT/9904058.[A3] S. Akbulut, , Book in preparation, available from , 2014.[Ak] A. Akhmedov
Small exotic 4-manifolds , Algebraic and Geometric Topology, 8(2008), 1781-1794.[AP] A. Akhmedov, B. D. Park,
Exotic 4-manifolds with small Euler characteristic ,Inventiones Mathematicae, 173 (2008), 209-223.[AY] S. Akbulut and K. Yasui,
Corks, Plugs and exotic structures ,Journal of G¨okova Geometry Topology, volume (2008), 40–82.[FPS] R. Fintushel, B. D. Park, R. J. Stern Reverse engineering small 4-manifolds , Alge-braic and Geometric Topology, 8 (2007), 2103-2116.[GS] R. E. Gompf and A.I. Stipsicz, 4 -manifolds and Kirby calculus , Graduate Studiesin Mathematics 20. American Mathematical Society, 1999.[BK] S. Baldridge, P. Kirk
A symplectic manifold homeomorphic but not diffeomorphicto CP CP , Geometry and Topology 12, (2008), 919-940. Department of Mathematics, Michigan State University, MI, 48824
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