How to Fold a Manifold. J. Scott Carter
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1 How to Fold a Manfold J. Scott Carter Unversty of South Alabama May 2012 KOOK Semnar
2 Outlne 1. Jont work wth Sech Kamada
3 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers
4 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust
5 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons
6 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons 5. Man Statements
7 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons 5. Man Statements 6. Why t s nterestng
8 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons 5. Man Statements 6. Why t s nterestng 7. Examples
9 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons 5. Man Statements 6. Why t s nterestng 7. Examples 8. Sketch of proof
10 Outlne 1. Jont work wth Sech Kamada 2. Thanks to organzers 3. Thanks to Branpool Trust 4. Defntons 5. Man Statements 6. Why t s nterestng 7. Examples 8. Sketch of proof
11 Let k = 1, 0, 1, or 2.
12 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2,
13 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k.
14 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k. When k = 1, L s empty.
15 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k. When k = 1, L s empty. When k = 0, L s an even number of ponts.
16 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k. When k = 1, L s empty. When k = 0, L s an even number of ponts. (orented)
17 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k. When k = 1, L s empty. When k = 0, L s an even number of ponts. (orented) When k = 1, L s a classcal lnk.
18 Let k = 1, 0, 1, or 2. Consder an orented lnk L k S k+2, and f : M k+2 S k+2 a smple (rregular) branched cover wth branch set L k. When k = 1, L s empty. When k = 0, L s an even number of ponts. (orented) When k = 1, L s a classcal lnk. When k = 2, we are consderng a lnked surface n the 4-sphere.
19 n n-1 n-1 regular pt. branch pt. 1 1 branch pt. regular pt.
20 Defnton A regular pt. y of an n fold smple branched cover has a Eucldean nbhd. N(y) such that f 1 (N(y)) has n-components: N 1,..., N n each homeom. to N(y).
21 Defnton A regular pt. y of an n fold smple branched cover has a Eucldean nbhd. N(y) such that f 1 (N(y)) has n-components: N 1,..., N n each homeom. to N(y). At a branch pont there are only (n 1) premages,
22 Defnton A regular pt. y of an n fold smple branched cover has a Eucldean nbhd. N(y) such that f 1 (N(y)) has n-components: N 1,..., N n each homeom. to N(y). At a branch pont there are only (n 1) premages, and one component of f 1 (N(y)) s of the form: (x 1, x 2,..., x k+2 ) (x 2 1 x 2 2, 2x 1 x 2,..., x k+2 )
23 Defnton A regular pt. y of an n fold smple branched cover has a Eucldean nbhd. N(y) such that f 1 (N(y)) has n-components: N 1,..., N n each homeom. to N(y). At a branch pont there are only (n 1) premages, and one component of f 1 (N(y)) s of the form: (x 1, x 2,..., x k+2 ) (x 2 1 x 2 2, 2x 1 x 2,..., x k+2 ) wth y the mage of 0.
24 Man Results Theorem Let k = 1, 0, 1, or 2. Let f : M k+2 S k+2 be an rregular smple branched cover of S k+2 branched along an embedded orented (poss. dsconn.) smooth sub-manfold L k S k+2. Then there s an mmerson f : M k+2 S k+2 D 2 such that the restrcton of the projecton onto the frst factor s a coverng map.
25 Such an mmerson wll be called an mmersed foldng. In ths case we wll say that the cover s folded. If f s, n fact, an embeddng, then t wll be called an embedded foldng.
26 Corollary Let k = 1, 0, 1, 2. Any compact orented (k + 2)-dmensonal manfold can be mmersed n R k+4.
27 Theorem Let k = 1, 0, 1, 2. Let L S k+2 denote any orentable compact smoothly embedded lnk. The 2-fold branched cover of S k+2 branched along L can be folded and embedded n S k+2 D 2.
28 Theorem (Hlden-Montesnos) Every closed connected orented 3-dmensonal manfold s obtaned as a smple rregular 2 or 3-fold branched cover of S 3 branched along some lnk.
29 Theorem (Ior-Pergalln) Every closed connected orented PL 4-dmensonal manfold can be obtaned as a smple rregular branched cover of S 4 branched along a (lnked) surface F S 4 wth of degree less than or equal to 5.
30 Remark The embedded foldngs are analogous to brads.
31 Remark The embedded foldngs are analogous to brads. In fact, they are hgher dmensonal brads.
32 Remark The embedded foldngs are analogous to brads. In fact, they are hgher dmensonal brads. And they are descrbed qute explctly as sequences of sequences of brad words.
33
34 In case k = 0, the smple branch pont s modeled upon the graph of z z 2.
35 In case k = 0, the smple branch pont s modeled upon the graph of z z 2. In the next slde, the 2-fold branched cover of a dsk (whch s an annulus) s llustrated.
36 In case k = 0, the smple branch pont s modeled upon the graph of z z 2. In the next slde, the 2-fold branched cover of a dsk (whch s an annulus) s llustrated. Smple branched covers of surfaces can all be bult from ths example.
37
38 Brad group B n = σ 1, σ 2,..., σ n 1 : σ σ j = σ j σ f j > 1; σ σ +1 σ = σ +1 σ σ +1 f = 1,... n 2
39 Permutaton group Σ n = τ 1, τ 2,..., τ n 1 : τ τ j = τ j τ f j > 1; τ τ +1 τ = τ +1 τ τ +1 f = 1,... n 2; τ 2 = 1, f = 1,..., n 1
40 A (permutaton) chart of degree n s a labeled fnte graph embedded n the 2-dsk D 2 wth three types of vertces:
41 A (permutaton) chart of degree n s a labeled fnte graph embedded n the 2-dsk D 2 wth three types of vertces: Whte vertex black vertex j 1< - j crossng
42 A (permutaton) chart of degree n s a labeled fnte graph embedded n the 2-dsk D 2 wth three types of vertces: Whte vertex black vertex j 1< - j crossng A brad chart ncludes orentatons on the edges.
43 Type II move
44 Black vertces
45 Crossng exchange
46 Whte vertces
47 Example b+ 0 b τ + 0 χ b 1 τ 3 τ 1 τ1 τ 2 b 3 τ + 0 b 1 τ 2 τ τ 2 τ 1 τ 2 W τ 1 τ 2 τ 1 b 2 τ 1 τ 1 II 1.
48
49
50 In the general case when k = 1
51 In the general case when k = 1 (branched covers of S 3 over a lnk,
52 In the general case when k = 1 (branched covers of S 3 over a lnk, the coverng wll be gven as a sequence of charts that are connected by moves of the followng type:
53 Chart moves +1 Empty dagram j j -j > j k j k -j >1; -k >1; j-k >1 j j > j j -j > j approprate ndces
54 For 2-fold branched covers, we use the followng moves.
55 1 H = or = 2 H Handle attachment empty. add 1-handle add 2-handle
56 IIb = a type II bubble move empty.
57 = A type II saddle move ) (.
58 CC = Candy-cane moves or,,,.
59 X = Exchange moves,, or,.
60 Z = Type II Zg-Zag
61 Lemma. Proof. 1-H IIs CC CC
62 3 tmes
63 [( )] 1 H = [( ) b+ 0 (σ1 1 ) b 1 ( )] = 1 H [( ) b+ 0 (σ1 1 ) b 1 ( ) b+ 0 (σ1 1 ) b 1 ( )] X = [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] IIs = [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) II 1 ( ) II+ 1 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] [( ) b+ 0 (σ 1 CC = CC = 1 ) b 1 ( ) II+ 1 (σ 1 σ 1 1 ) b 2 [( ) b+ 0 (σ1 1 ) b 1 ( ) b+ 0 (σ1 1 ) b 1 ( )] (σ 1 ) b 1 ( )] X =
64 [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] IIs = [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) II 1 ( ) II+ 1 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] [( ) b+ 0 (σ 1 CC = CC = 1 ) b 1 ( ) II+ 1 (σ 1 σ 1 1 ) b 2 [( ) b+ 0 (σ1 1 ) b 1 ( ) b+ 0 (σ1 1 ) b 1 ( )] (σ 1 ) b 1 ( )] X = [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] IIs = [( ) b+ 0 (σ1 1 ) b+ 0 (σ 1 σ1 1 ) II 1 ( ) II+ 1 (σ 1 σ1 1 ) b 2 (σ 1 ) b 1 ( )] CC =
65 [( ) b+ 0 (σ 1 CC = 1 ) b 1 ( ) II+ 1 (σ 1 σ 1 1 ) b 2 (σ 1 ) b 1 ( )] [( ) b+ 0 (σ1 1 ) b 1 ( ) b+ 0 (σ 1 ) b 1 ( )] =. 2 H [( ) b+ (σ 1 ) b ( )] [( )]. 2H =
66
67
68 Closng remarks The 3-fold branched cover of S 3 over a knot may not always be a folded embeddng.
69 Closng remarks The 3-fold branched cover of S 3 over a knot may not always be a folded embeddng. For example, 7 4 s 3-colorable, but the foldng has a smple closed curve of double ponts.
70 Closng remarks The 3-fold branched cover of S 3 over a knot may not always be a folded embeddng. For example, 7 4 s 3-colorable, but the foldng has a smple closed curve of double ponts. Nonetheless, the cover s a 3-sphere. It s not braded over 7 4. :(
71 Closng remarks The 3-fold branched cover of S 3 over a knot may not always be a folded embeddng. For example, 7 4 s 3-colorable, but the foldng has a smple closed curve of double ponts. Nonetheless, the cover s a 3-sphere. It s not braded over 7 4. :( For 2-fold branched covers of S 4 over orentable manfolds, we can always fnd embedded foldngs.
72 Closng remarks The 3-fold branched cover of S 3 over a knot may not always be a folded embeddng. For example, 7 4 s 3-colorable, but the foldng has a smple closed curve of double ponts. Nonetheless, the cover s a 3-sphere. It s not braded over 7 4. :( For 2-fold branched covers of S 4 over orentable manfolds, we can always fnd embedded foldngs. Menton CP 2 and the 3-fold branched cover of the 2-twst-spun trefol
73
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