CS 6170: Computational Topology, Spring 2019 Lecture 09

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1 CS 6170: Computtionl Topology, Spring 2019 Lecture 09 Topologicl Dt Anlysis for Dt Scientists Dr. Bei Wng School of Computing Scientific Computing nd Imging Institute (SCI) University of Uth Feb 5, 2019

2 2-dimensionl Mnifold Book Chpter A.II

3 2-mnifold without boundry A 2-mnifold without boundry is topologicl spce M whose points ll lie in open disks. Intuitively, this mens tht M loclly looks like plne. We get 2-mnifold with boundry by removing open disks from 2-mnifolds without boundry.

4 Exmples of 2-mnifolds <ltexit sh1_bse64="nvbij9sjva60v68j48k3ihzntw=">aaab9xicbvbntwixfhylx4hfqecvjwdiiexy0zmh8eiro4ajlkrbutdqdjdtv0m2/a8vhjtgq//fm//gluxbwumtgbey5toehomjet+o4w19y3nrej2wd3b/+gfhju1lgicg2riefqicccizpyzdd6uoskbybp51gcp35nueqnivkvzng1bd4jfnicdzw6ld7aptxekr3s369oihx3jo7b1olxk4qkkm5kh/1hhfjbjwgckx113nj46dyguy4nzv6iyxjhm8ol1ljrzu++k89qydwwwiwkjzjw2q783uiy0noratmyh9bkxif953csel37kzjwyksniujhwzckuvycgtffi+nqstbszwrezy4wjsuwvbane8pdxsbte89yd1uvnk7yoopwaqdwdh5cqanuoaktikdggv7hzxlyxpx352mxwndynwp4a+fzb6dzke0=</ltexit> <ltexit sh1_bse64="nvbij9sjva60v68j48k3ihzntw=">aaab9xicbvbntwixfhylx4hfqecvjwdiiexy0zmh8eiro4ajlkrbutdqdjdtv0m2/a8vhjtgq//fm//gluxbwumtgbey5toehomjet+o4w19y3nrej2wd3b/+gfhju1lgicg2riefqicccizpyzdd6uoskbybp51gcp35nueqnivkvzng1bd4jfnicdzw6ld7aptxekr3s369oihx3jo7b1olxk4qkkm5kh/1hhfjbjwgckx113nj46dyguy4nzv6iyxjhm8ol1ljrzu++k89qydwwwiwkjzjw2q783uiy0noratmyh9bkxif953csel37kzjwyksniujhwzckuvycgtffi+nqstbszwrezy4wjsuwvbane8pdxsbte89yd1uvnk7yoopwaqdwdh5cqanuoaktikdggv7hzxlyxpx352mxwndynwp4a+fzb6dzke0=</ltexit> <ltexit sh1_bse64="nvbij9sjva60v68j48k3ihzntw=">aaab9xicbvbntwixfhylx4hfqecvjwdiiexy0zmh8eiro4ajlkrbutdqdjdtv0m2/a8vhjtgq//fm//gluxbwumtgbey5toehomjet+o4w19y3nrej2wd3b/+gfhju1lgicg2riefqicccizpyzdd6uoskbybp51gcp35nueqnivkvzng1bd4jfnicdzw6ld7aptxekr3s369oihx3jo7b1olxk4qkkm5kh/1hhfjbjwgckx113nj46dyguy4nzv6iyxjhm8ol1ljrzu++k89qydwwwiwkjzjw2q783uiy0noratmyh9bkxif953csel37kzjwyksniujhwzckuvycgtffi+nqstbszwrezy4wjsuwvbane8pdxsbte89yd1uvnk7yoopwaqdwdh5cqanuoaktikdggv7hzxlyxpx352mxwndynwp4a+fzb6dzke0=</ltexit> <ltexit sh1_bse64="nvbij9sjva60v68j48k3ihzntw=">aaab9xicbvbntwixfhylx4hfqecvjwdiiexy0zmh8eiro4ajlkrbutdqdjdtv0m2/a8vhjtgq//fm//gluxbwumtgbey5toehomjet+o4w19y3nrej2wd3b/+gfhju1lgicg2riefqicccizpyzdd6uoskbybp51gcp35nueqnivkvzng1bd4jfnicdzw6ld7aptxekr3s369oihx3jo7b1olxk4qkkm5kh/1hhfjbjwgckx113nj46dyguy4nzv6iyxjhm8ol1ljrzu++k89qydwwwiwkjzjw2q783uiy0noratmyh9bkxif953csel37kzjwyksniujhwzckuvycgtffi+nqstbszwrezy4wjsuwvbane8pdxsbte89yd1uvnk7yoopwaqdwdh5cqanuoaktikdggv7hzxlyxpx352mxwndynwp4a+fzb6dzke0=</ltexit> Top: 2-mnifold without boundry Bottom: 2-mnifold with boundry Möbius strip: non-orientble mnifold; 2 sides loclly, 1 side globlly. Möbius strip: n nt will trvel ll surfce re Möbius strip: its boundry is single circle Quiz: wht hppens if you cut Mobius strip long its center line? Sphere S 2 Torus Double torus T 2 #T 2 T 2 Disk Cylinder Mobius strip

5 Orientbility If ll closed curves in 2-mnifold re orienttion-preserving, then the 2-mnifold is orientble. Creting compct 2-mnifolds using polygonl schem. M is compct if for every covering of M by open sets, clled n open cover, we cn find finite number of the sets tht cover M. A subset of Eucliden spce is compct if it is closed nd bounded (i.e., contined in bll of finite rdius).

6 Clssifiction Theorem (Clssifiction theorem for compct 2-mnifolds) The two infinite fmilies S 2, T 2, T 2 #T 2,, nd P 2, P 2 #P 2,, exhust the compct 2-mnifolds without boundry. (Edelsbrunner nd Hrer, 2010, Pge 29)

7 Clssifiction Any connected closed surfce is homeomorphic to some member of one of these three fmilies: The sphere The connected sum of g tori, for g 1 The connected sum of k rel projective plnes, for k 1.

8 Polygonl schem <ltexit sh1_bse64="nd9xbv395ffxeguie+dqmhcttrg=">aaab9xicbvdltgixfl2dl8qx6tjni5i4ijnsdgvi3ljerb4jdkrtotdqdiztr0mm/icbfxrj1n9x59/ygvkoejimj+fcm3t6gpgzbvz32ylsbg5t7xr3s3v7b4dh5eotto4srwilrdxs3qbrypmklcmmp91yuswctjvb9dbzo49urbjbzolqs/wwlkqewysnkj2bttieib80g9oixx3jq7afonxk4qkkm5lh/1rxfjbjwgckx1z3nj46dyguy4nzf6iyxjlm8pj1ljrzu++ki9rxdwgwewkjzjw1ql83uiy0nonatmyh9qxif95vcse137kzjwyksnyujhwzckuvybgtffi+mwstbszwrgzyiwjsuwvbane6pfxsbte89yd1+vng7yoopwbudwcr5cqqpuoaktikdggv7hzxlyxpx352m5wndynvp4a+fzb6lbkeo=</ltexit> <ltexit sh1_bse64="nd9xbv395ffxeguie+dqmhcttrg=">aaab9xicbvdltgixfl2dl8qx6tjni5i4ijnsdgvi3ljerb4jdkrtotdqdiztr0mm/icbfxrj1n9x59/ygvkoejimj+fcm3t6gpgzbvz32ylsbg5t7xr3s3v7b4dh5eotto4srwilrdxs3qbrypmklcmmp91yuswctjvb9dbzo49urbjbzolqs/wwlkqewysnkj2bttieib80g9oixx3jq7afonxk4qkkm5lh/1rxfjbjwgckx1z3nj46dyguy4nzf6iyxjlm8pj1ljrzu++ki9rxdwgwewkjzjw1ql83uiy0nonatmyh9qxif95vcse137kzjwyksnyujhwzckuvybgtffi+mwstbszwrgzyiwjsuwvbane6pfxsbte89yd1+vng7yoopwbudwcr5cqqpuoaktikdggv7hzxlyxpx352m5wndynvp4a+fzb6lbkeo=</ltexit> <ltexit sh1_bse64="nd9xbv395ffxeguie+dqmhcttrg=">aaab9xicbvdltgixfl2dl8qx6tjni5i4ijnsdgvi3ljerb4jdkrtotdqdiztr0mm/icbfxrj1n9x59/ygvkoejimj+fcm3t6gpgzbvz32ylsbg5t7xr3s3v7b4dh5eotto4srwilrdxs3qbrypmklcmmp91yuswctjvb9dbzo49urbjbzolqs/wwlkqewysnkj2bttieib80g9oixx3jq7afonxk4qkkm5lh/1rxfjbjwgckx1z3nj46dyguy4nzf6iyxjlm8pj1ljrzu++ki9rxdwgwewkjzjw1ql83uiy0nonatmyh9qxif95vcse137kzjwyksnyujhwzckuvybgtffi+mwstbszwrgzyiwjsuwvbane6pfxsbte89yd1+vng7yoopwbudwcr5cqqpuoaktikdggv7hzxlyxpx352m5wndynvp4a+fzb6lbkeo=</ltexit> <ltexit sh1_bse64="nd9xbv395ffxeguie+dqmhcttrg=">aaab9xicbvdltgixfl2dl8qx6tjni5i4ijnsdgvi3ljerb4jdkrtotdqdiztr0mm/icbfxrj1n9x59/ygvkoejimj+fcm3t6gpgzbvz32ylsbg5t7xr3s3v7b4dh5eotto4srwilrdxs3qbrypmklcmmp91yuswctjvb9dbzo49urbjbzolqs/wwlkqewysnkj2bttieib80g9oixx3jq7afonxk4qkkm5lh/1rxfjbjwgckx1z3nj46dyguy4nzf6iyxjlm8pj1ljrzu++ki9rxdwgwewkjzjw1ql83uiy0nonatmyh9qxif95vcse137kzjwyksnyujhwzckuvybgtffi+mwstbszwrgzyiwjsuwvbane6pfxsbte89yd1+vng7yoopwbudwcr5cqqpuoaktikdggv7hzxlyxpx352m5wndynvp4a+fzb6lbkeo=</ltexit> b b Torus b b Sphere b P 2 Projective Plne b b Klein bottle b Projective plne: glue disk to Möbius strip Klein bottle: glue 2 Möbius strips together

9 Klein bottle: non-orientble surfce Lter: show up in dt nlysis of nturl imge ptches Crlsson et l. (2008).

10 Betti numbers β i of 2-mnifolds β 0 β 1 β 2 circle sphere torus projective plne Klein bottle hole torus g-hole torus 1 2g 1

11 Genus The genus of connected, orientble surfce is n integer representing the mximum number of cuttings long non-intersecting closed simple curves without disconnecting the resulting mnifold. lso for further reding

12 Computing Homology Book Chpter B.IV.

13 Reduction of boundry mtrix Let p be the p-th boundry mtrix After reduction (row nd column opertion), p turns out to be mtrix N p in Smith Norml Form β p = rnk Z p rnk B p

14 Exmple: tringultion of circle 0 = N 0 is 1 3 mtrix with ll 0 entries. rnk C 0 = rnk Z 0 = 3; rnk Z 1 = 1, rnk B 0 = 2 β 0 = rnk Z 0 rnk B 0 = 3 2 = 1 β 1 = rnk Z 1 rnk B 1 = 1 0 = 1

15 Tke home exercise The following simplicil complex contins 4 vertices, 6 edges, 3 tringles. Compute its Betti numbers: β 0 = 1, β 1 = 0, β 2 = 1. b c d

16 References I Crlsson, G., Ishkhnov, T., De Silv, V., nd Zomorodin, A. (2008). On the locl behvior of spces of nturl imges. Interntionl journl of computer vision, 76(1):1 12. Edelsbrunner, H. nd Hrer, J. (2010). Computtionl Topology: An Introduction. Americn Mthemticl Society, Providence, RI, USA.