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

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. Pingback: two and three dimensional manifolds « juanmarqz

  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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