# elementary 3-manifolds

Strangely, there are very few places that offer simple types or simple constructions of these topological beings. Sometimes innocent guessing make us to think that only $\mathbb{R}^3$ is the only space we need to appreciate all the available 3d-complexity.

Today there are: “obvious ones”

1. EIGHT WITH GEOMETRIC STRUCTURE
2. $\mathbb{R}^3$
3. $\mathbb{S}^3$
4. $\mathbb{H}^3$
5. $S^2\times\mathbb{R}$
6. $\mathbb{H}^2\times\mathbb{R}$
7. $\widetilde{SL_2(\mathbb{R})}$ the universal cover of $SL_2(\mathbb{R})$
8. ${\rm{nil}}$
9. ${\rm{sol}}$
10. TRIVIALS
11. $\mathbb{R}P^3$
12. $\mathbb{R}^2\times S^1$
13. $S^2\times S^1$
14. $O_g\times\mathbb{R}$
15. $N_k\times\mathbb{R}$
16. $O_g\times S^1$
17. $N_k\times S^1$
18. $F\times S^1$, where $F$ is any surface (closed or punctured)
19. $G\times S^1$, where $G$ is a 2-orbifold
20. COMPLEMENTS
21. knot complements in $\mathbb{R}^3$
22. knot complements in $S^3$
23. link complements in $\mathbb{R}^3$
24. GENUS ONE
25. lens spaces
26. SURFACE BUNDLES
27. $O_g\subset E\to S^1$
28. $N_k\subset E\to S^1$
29. $F\subset E\to S^1$
30. $G\subset E\to S^1$
31. $D\subset D\times S^1\to S^1$
32. I-BUNDLES
33. $I\subset E\to O_g$
34. $I\subset E\to N_k$
35. $I\subset E\to F$
36. $I\subset E\to G$
37. CIRCLE BUNDLES
38. $S^1\subset E\to O_g$
39. $S^1\subset E\to N_k$
40. $S^1\subset E\to F$
41. $S^1\subset E\to G$
42. MORE COMPLEMENTS
43. knots and link complements in them and in  others
44. Open book decompositions
45. OTHER BASICS
46. $M\ddot{o}\subset E\to S^1$
47. $A\subset E\to S^1$
48. $N_2\times I$
49. $N_2\stackrel{\sim}\times I^O$
50. $N_2\stackrel{\sim}\times I^{NO}$
51. SEIFERT FIBER SPACES and SEIFERT-SCOTT
52. $(Oo,g|(1,b))$
53. $(Oo,g|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
54. $(On,k|(1,b))$
55. $(On,k|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
56. $(No,g|(1,b))$
57. $(No,g|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
58. $(NnI,k|(1,b))$
59. $(NnI,k|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
60. $(NnI\!I,k|(1,b))$
61. $(NnI\!I,k|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
62. $(NnI\!\!I\!\!I,k|(1,b))$
63. $(NnI\!\!I\!\!I,k|(1,b),(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)$
64. ORLIK-RAYMOND-FINTUSHEL
65. $\{b;(\epsilon,g,h,t);\quad\}$
66. $\{b;(\epsilon,g,h,t);(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)\}$
67. $\{b;(\epsilon,g,(\bar{h},k_1),(t,k_2));(\alpha_1,\beta_1),...,(\alpha_r,\beta_r)\}$

twisted I-bundle over the mobius strip, is, a solid torus but fibered 2:1

—————

Tools

• irreducibility
• $\mathbb{R}P^2$-irreducibility
• Milnor-Wallace-Kneser
• $\pi_1(M_1\#\cdots\#M_r)=\pi_1(M_r)\times\cdots\times\pi_1(M_r)$
• Eilenberg-Maclane $K(G,k)$-spaces
• incompressible
• Dehn’s Lemma, Loop’s, Sphere’s, Annulus’, Torus’ theorems
• Haken
• JSJ-decomposition
• Geometrization theorem (Poincarè-Thurston-Hamilton-Perelman)

Splitting and Non-orientable

• Heegaard splittings $M=H_1\cup_{f:\partial}H_2$
• $h(M)=\min_{H_i} g(H_i)$
• characteristic classes $\mod2$
• Bockstein morphism $\beta: H^1(M:\mathbb{Z}_2)\to H^2(M:\mathbb{Z}_2)$
• Poincarè duality
• Singhof fillings $M=V_1\cup V_2\cup V_3$, into  3 orientable handlebodies
• splitting type $st(M)=(gV_i,gV_j,gV_k)$ with  $gV_i\le gV_j\le gV_k$
• tri-genus ${\rm{trig}}(M)=\min st(M)$
• $w_1(M)=0$ then ${\rm{trig}}(M)=(0,h(M),h(M))$
• $w_1(M)\neq0$ but $\beta(w_1(M))=0$ then ${\rm{trig}}M)=(0,2g,k)$
• $w_1(M)\neq0$ but $\beta(w_1(M))\neq0$ then ${\rm{trig}}M)=(0,2g-1,k)$ where $g={\rm{SW\!\!- \!\!g}}(M)$ the Stiefel-Whitney genus of $M$ (i.e the minimal genus of a Stiefel-Whitney surface in $M$)

Spines and complexity

Applications

Prospective

3 and 4 manifolds

## REFERENCES:

### 7 responses to “elementary 3-manifolds”

1. peace&love

could you tell how many and how are the $N_3$-bundles over the circle?

2. km qit

http://hep.physics.indiana.edu/~tsulanke/graphs/surftri/counts.txt

for counts in triagulations of surfaces…

3. km-qit

${\cal{MCG}}(S^1\times S^1)=GL_2(\mathbb{Z})$

$\#\{ S^1\times S^1\subset E_{\varphi}\to S^1\}=7$ when $\varphi$ is periodic

4. km-qit

${\cal{MCG}}(N_2)=\langle t,y|t^2=y^2=e,ty=yt\rangle=\mathbb{Z}_2\oplus\mathbb{Z}_2$

$\#\{ N_2\subset E_{\varphi}\to S ^1\}=4$

—-

$E_1=N_2\times S^1=(No,1|0)=(NnI,2|0)$

$h(E_1)=3$

$\beta(w_1(E_1))=0$

${\rm{trig}}(E_1)=(0,2,2)$

—-

$E_t=N_2\times_t S^1=(No,1|1)=(NnI,2|1)$

$h(E_t)=2$

$\beta(w_1(E_t))=0$

${\rm{trig}}(E_t)=(0,2,2)$

—-

$E_y=N_2\times_y S^1=(NnI\!I,2|0)$

$h(E_y)=3$

$\beta(w_1(E_y))\neq0$

${\rm{trig}}(E_y)=(1,1,1)$

—-

$E_{ty}=N_2\times_{ty} S^1=(NnI\!I,2|1)$

$h(E_{ty})=2$

$\beta(w_1(E_{ty}))\neq0$

${\rm{trig}}(E_{ty})=(1,1,1)$

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