INTRODUCTION TO SIMPLICIAL COMPLEXES

Transcription

1 INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min gol of this ctivity is to lern how to construct certin topologicl invrints of different objects using simplicil complexes. These re visul, mthemticl structures tht represent shpes we know well but tht we re ble to perform computtions on. To outline, we will be (1) Discussing simplices, these re the building blocks to crete our complexes. We ll be exploring how to define, construct, nd write down these structures in different wys. (2) We then extend to building nd simplicil complexes. This will tke some tinkering nd definitions, nd ctivities to get comfortble. (3) Then we begin relting these complexes to structures we know. We will look t objects nd see how they cn be pproximted from simplicl complex. We show how we cn use the gluing opertion to crete objects from our simplicil complexes. (4) We introduce the boundry opertor,, which you cn use to mthemticlly compute the boundry of given surfce or object (you cn show, for exmple, tht the boundry of bll is most definitely sphere!). 1. A SIMPLEX An n-simplex is geometric object with (n + 1) vertices which lives in n n-dimensionl spce (nd cnnot fit in ny spce of smller dimension). The vertices of the simplex generte the simplex through simple geometric construction which we illustrte below. The ide is esy: one vertex genertes point, two vertices generte segment (by connecting the two points), three vertices generte tringle (by connecting every pir of points with segments nd filling the spce in between) nd so on. Notice how (n + 1) vertices re needed to generte n object of dimension n. Also note tht becuse we wnt n n-simplex to be n object of dimension n, bit of cre must be exercised in the choice of its (n + 1) generting vertices. For exmple, three points which belong to the sme line hve no hope to generte 2-dimensionl object! We now outline the steps for building n n-simplex (for n = 0, 1, 2, 3). These re the only simplexes we cn visulize. The construction generlizes to simplices of bigger dimension, but we will need to rely on our imgintion to picture the ctul geometric objects. Using precise mthemticl nottion fcilittes the bstrction nd helps you think bout higher dimensionl structures or concepts tht often cn t even be visulized. Definition 1.1 (n-simplexes, for n = 0, 1, 2, 3). Strt out with the 3-D spce nd drw three coordinte xes. Your xes do not necessrily hve to be perpendiculr to ech other, just mke sure they do not crush into plne. We will be denoting generl simplex by σ nd, in prticulr, n n-simplex by σ n. 0-simplex ( simplex generted by one point, ) A 0-simplex is point; for exmple, the origin or nother point in the coordinte xis σ 0 =. 1

2 2 CASEY KELLEHER AND ALESSANDRA PANTANO Step 1 build For simplicity, we will lwys tke to be t the origin of our coordinte xes. 1-simplex ( simplex, p 1 generted by two points, nd p 1 ) A 1-symplex is line segment (including its end-points). To build one, tke the origin nd 1 other point which lies on coordinte xis. This construction, produces two 0-subsimplices. Next, connect the two points to get your 1-simplex σ 1 =, p 1. Step 1 Step 2 build, p 1 build, p 1 p 1 p 1 2-simplex ( simplex, p 1, p 2 generted by three points,, p 1, p 2 ) A 2-simplex is solid tringle (including its border). To build one, tke the origin nd 2 other points which lie on two different coordinte xis. So fr, this gives you three 0-subsimplices. Next, connect ll possible pirs of two points, to get three 1-subsimplices. Finlly, fill in the resulting tringle to obtin your 2-simplex σ 2 =, p 1, p 2. Step 1 Step 2 Step 3 build, p 1, p 2 build, p 1,, p 2, p 1, p 2 build, p 1, p 2 p 1 p 1 p 1 p 2 p 2 p 2 3-simplex ( simplex, p 1, p 2, p 4 generted by three points,, p 1, p 2, p 4 )

3 INTRODUCTION TO SIMPLICIAL COMPLEXES 3 Tsk 1. Fill in the blnks nd complete the pictures below.: A 3-simplex is solid tetrhedron (including its border). To build one, tke the origin nd other points which lie on different coordinte xis. This construction produces. Next, connect ll possible pirs of two points, to get 1- subsimplices. The next step is to, to obtin. Finlly, fill in the resulting to obtin your -simplex, σ 3. Step 1 Step 2 Step 3 Step 4 Definition 1.2 (n-simplex). To construct n n-simplex for n > 3, iterte this construction. A simplex, mthemticlly, doesn t hve ny fixed shpe or size, or orienttion. In prticulr, the following (1) Rigid motions. (Rotte, trnslte, dilte) We cn move or rotte the simplex to nywhere we desire, nd it still counts s the sme simplex. (2) Stretch. We cn stretch out generting points wy from ech other (nd chnge the connected structures too). However, you cnnot crush simplices- you cnnot turn n n-simplex into n (n 1)-simplex by deforming it.

4 4 CASEY KELLEHER AND ALESSANDRA PANTANO A remrks bout simplices. The order in which we list the vertices generting simplex does not mtter, for exmple:, b, c = c,, b = b, c,. This fct holds true for ny n-simplex (lter, we ll be dding orienttion to these structures, then the order will mtter!). Tsk 2. Wht do the following structures need to become simplexes? Wht will be the dimension of ech resulting simplex? We hve included xes for your convenience!. First lbel the points nd write out the components using the nottion we hve shown bove to help see wht you re missing. Definition 1.3 (fce). Let σ =, p 1,..., p n be n n-dimensionl simplex. A fce of σ is subsimplex of σ, nmely, the simplex generted by subset of the vertices of σ. To get fce of σ of dimension m n, choose m + 1 points mong, p 1,..., p n nd tke the corresponding simplex. Tsk 3. Count the number of m-simplexes needed to construct n n-simplex. An exmple is provided.

5 INTRODUCTION TO SIMPLICIAL COMPLEXES 5 # of m-simplexes contined in n n-simplex m = 0 m = 1 m = 2 m = 3 m = 4 n = n = n = n = 3 n = 4 Question 1.4 (Chllenge question). Cn you think of generl rule for these formuls? Exmple 1.5. p 1, p 3 is 1-dimensionl fce of, p 1, p 2, p 3 : p 1 p 3 Similrly,, p 3, p 2 is 2-dimensionl fce of, p 1, p 2, p 3 : p 2 p 1 p 3 p 2

6 6 CASEY KELLEHER AND ALESSANDRA PANTANO 2. A SIMPLICIAL COMPLEX Definition 2.1 (simplicil complex). A simplicil complex K is collection of simplices such tht (1) If K contins simplex σ, then K lso contins every fce of σ. (2) If two simplices in K intersect, then their intersection is fce of ech of them. Remrk 2.2. When we write simplex K, we use set nottion (tht is, squiggly brckets contining ll of the simplexes which re included in the simplicil complex: K = {σ 1,..., σ n } ). In comprison to simplex, we think bout simplicil complex s set with visul representtion. The simplex is building block to crete the simplicil complex. Exmple 2.3. Here re some exmples (nd nonexmple) of simplex, including both the digrm nd set nottion. Ex 1 K = {, p 1, p 2, p 3, p 1, p 2, p 3,, p 2, p 3,, p 1, p 3,, p 1, p 2,, p 1,, p 2,, p 3, p 1, p 2, p 1, p 3, p 2, p 3,, p 1, p 2, p 3. p 1 p 3 p 2 Ex 2 K = {, p 1, p 2,, p 1,, p 2, p 1, p 2, p 2, p 3, p 2, p 4, p 3, p 4, p 4, p 5,, p 1, p 2, p 3, p 4, p 5. p 3 p 4 p 5 p 2 Ex 3 p 1 These re few NON-EXAMPLES, which we will denote by J (they ren t simplicil complexes). Tsk 4. In the spces below explin why these sets J re not simplicil complexes nd how you would fix them (Drw J first, then dd the pproprite simplex pieces to turn J into simplicil complex K). For simplicity, we will use distinct letters from the lphbet to lbel the points. Check with your neighbor on wht you drew- do these look correct?

7 INTRODUCTION TO SIMPLICIAL COMPLEXES 7 ( Exmple) J = {, b, c,, b,, c, d, e } Solution: c b e J K d K = {, b, c,, b,, c, b, c, d, e,, b, c, d, e } Note tht this corrected simplicil complex K hs two disjoint pieces. This is oky! It still stisfies the definition of simplicil complex. J = {, b, c, b, c, c, d,, b, c } Solution: K = J = {, b, c, d, d, e, f } Solution: K = 2.1. Skeletons. Now tht we hve some exmples of simplexes, we re discuss how to sort nd consider the vrious pieces which constitute the simplex. In prticulr, their skeletons!. Definition 2.4 (p-skeleton). The p-skeleton of simplicil complex K is denoted by K (p) nd is the set of ll of the simplices in K of dimension p or less. Exmple 2.5. We give lists of the skeletons corresponding to Ex 1 nd Ex 2 bove. Fill in the gps.

8 8 CASEY KELLEHER AND ALESSANDRA PANTANO p 1 p 3 p 2 Ex 1 K (0) = {, p 1, p 2, p 3 } (the vertices) K (1) = { } (the vertices nd the edges) K (2) = {, p 1, p 2, p 3, p 1, p 2, p 3,, p 2, p 3,, p 1, p 3,, p 1, p 2,, p 1,, p 2,, p 3, p 1, p 2, p 1, p 3, p 2, p 3,, p 1, p 2, p 3 } (the vertices, the edges nd the tringles) K (3) = K (the vertices, the edges, the tringles nd the tetrhedron). Tsk 5. Drw K (n) for ech n listed. Does the nme skeleton mke sense? K (0) K (1) K (2) K (3) p 3 p 4 p 5 p 2 p 1 Ex 2 List the elements in ech set: K (0) = { } K (1) = { } K (2) = { } Tsk 6. Drw K (n) for ech n listed.

9 INTRODUCTION TO SIMPLICIAL COMPLEXES 9 K (0) K (1) K (2) K (3) Question 2.6. Argue whether ech of these clims is true or flse. For ll n, K (n) K (n+1). If n = dim(k), K (n) = K. If n > dim(k), K (n) = (the empty set).

10 10 CASEY KELLEHER AND ALESSANDRA PANTANO 2.2. Model complexes. Tsk 7. Crete simplicil complexes which model the following rel life objects, including both (1) lbeled digrm nd (2) set nottion, s we did in exmples bove. Object chir bottle blloon Digrm Set nottion

11 INTRODUCTION TO SIMPLICIAL COMPLEXES 11 Question 2.7 (Chllenge Question). How mny fces would you need for comb with n bristles? Or brush with n bristles? (Here we hve only dded few bristles to the comb nd one to the brush, these re colored in purple)

12 12 CASEY KELLEHER AND ALESSANDRA PANTANO 3. GLUING Tsk 8. Using the grids on the next pge, crete the following objects (1) Cylinder (2) Möbius strip (3) Torus

13 INTRODUCTION TO SIMPLICIAL COMPLEXES 13 Tsk 9 (Chllenge question). The structure you will get from this is four dimensionl surfce is the Klein bottle. Using this grid, drw sketch of how you think this will look (you cn drw it with sides intersecting).

14 14 CASEY KELLEHER AND ALESSANDRA PANTANO d e b f g c c h i b e d Sketch: ORIENTATION Now we re going to dd orienttion to our simplicil complex. It my seem strnge t first, but it will help lter with both gluing nd ppropritely defining out boundry opertor.

15 INTRODUCTION TO SIMPLICIAL COMPLEXES 15 Definition 3.1 (oriented simplex). σ is n oriented p-simplex if it is p-simplex nd hs fixed orienttion (tht is, the order of the vertexes is fixed). To denote n oriented simplex, we will use brckets [ ] insted of symbols round the generting vertices. These oriented simplices come with the property tht [ p0, p 1,..., p i,..., p j,... p n 1, p n ] = [ p0, p 1,..., p j,..., p i,... p n 1, p n ]. (Switching two vertices introduces minus sign) To drw oriented simplices, we will only consider n-simplices for n [1, 2, 3] Exmple 3.2. Here re pictures of 1-oriented simplices, nd 2-oriented simplices. [, b] [b, ] b b [, b, c] [, c, b] c c b b Finlly, for 3 simplex, we would drw [, b, c, d] s follows. c d b Tsk 10. There re 6 orderings of the vertices of tringle. 2-simplices: [, c, b] nd [, b, c]. For exmple: [c,, b] = [, c, b] = [, b, c]. However, there re only 2 oriented Similrly, tetrhedron would only hve two orienttions: [, c, b, d] nd [, b, c, d]. Which of these two is equivlent to [c, d,, b]?

16 16 CASEY KELLEHER AND ALESSANDRA PANTANO Definition 3.3 (p-chin). We cn dd p-simplices with integer coefficients to form chin. Remrk 3.4. In wy, you cn think or n oriented simplex s representing n ction. [, b] represents moving from to b, so if you move from to b nd then from b to, then the 2-chin representing this is the sme s dding the movements: [, b] + [b, ] = [, b] [, b] = 0. Which is sying, mthemticlly, you didn t get nywhere! Similrly, 2-simplices re like turns in different directions, so dding [, b, c] to [, c, b] lso vnishes. However, these keep trck of how mny totl spins mde (so even if you dd up mny turns in the sme direction they never cncel out ech other). Definition 3.5 (Oriented simplicil complex). An oriented simplicil complex is simplicil complex where ll of its chins re oriented. If simplicil complex K hs oriented subsimplices, it is n oriented simplicil complex nd denoted by K. Exmple 3.6. Here re some oriented simplicil complexes. Note tht we hve tken our nonoriented exmples from before nd put n orienttion on them. { [, p 1, p 2 ], [, p 1 ], [, p 2 ], [p 1, p 2 ], [p 2, p 3 ], [p 2, p 4 ], [p 3, p 4 ], [p 4, p 5 ], K = [ ], [p 1 ], [p 2 ], [p 3 ], [p 4 ], [p 5 ]. A key point is it doesn t mtter which wy you orient ech subsimplex, s long s you orient them! Now remember, the direction tht you orient them does effect wht they re equl to. For exmple, (we ve underlined the chnge): { K [, p 1, p 2 ], [p 1, ], [, p 2 ], [p 1, p 2 ], [p 2, p 3 ], [p 2, p 4 ], [p 3, p 4 ], [p 4, p 5 ], = [ ], [p 1 ], [p 2 ], [p 3 ], [p 4 ], [p 5 ]. As only simplicil complexes, K is equl to K, but s oriented simplicil complexes they re not equl. Definition 3.7 (Simplicil complex chins). ( ) The set of ll possible p-chins generted from n oriented simplicil complex K is denoted by C p K. Exmple 3.8. Here re some exmples of p-chins in K: (1-chin): c 1 = [, p 1 ] + 3 [p 2, p 3 ], (1-chin): c 1 = 2 [p 2, p 4 ] + 3 [, p 2 ], (2-chin): c 2 = 4 [, p 1, p 2 ]. ( ) ( Question 3.9. For the two exmples bove, how does C p K compre to C p K )? (Hint: how does negtive sign help?). Cn you sy something in generl bout oriented simplicil complexes formed from the sme simplicil complexes?

17 INTRODUCTION TO SIMPLICIAL COMPLEXES 17 BOUNDARY Now we cn finlly define the boundry opertor. Definition 3.10 (Boundry opertor). cts on oriented p-simplices s follows [,..., p n ] = n ( 1) n [,..., ˆp i,..., p n ], i=1 [p i ] = 0. (Here the ht p i mens we tke p i out from the simplex.) The min point is we go through writing the (p 1)-simplices where the ith entry is tken out, but with lternting signs. Since this definition cn be confusing, let s puse to give n exmple. Exmple [, b, c] = ( 1) 0 [â, b, c] + ( 1) 1 [, ˆb, c ] + ( 1) 2 [, b, ĉ] = [b, c] [, c] + [, b] = [b, c] + [c, ] + [, b] Here is visul, you cn visully see how it literlly computed the boundry of the shpe! [, b, c] [, b, c] c c b b Tsk 11. Compute [, b, c, d] using the formul, then sketch your results s we did in the exmple (use the spce in the next pge). Definition To extend to p-chin, we just mke it hit ll of the simplices nd ignore signs nd constnts in between. (c σ) = c σ, ( σ 1 + σ 2 ) = σ 1 + σ 2, where c is sclr, nd σ is simplex where σ 1, σ 2 re two simplices. Exmple 3.13 (Boundry of chin). Consider the bowtie below, given by if we compute the boundry we hve c = [, b, c] + [c, d, e] ([, b, c] + [c, d, e]) = [, b, c] + [c, d, e] = [b, c] [, c] + [, b] + [d, e] [c, e] + [d, e].

18 18 CASEY KELLEHER AND ALESSANDRA PANTANO Now if we compute gin using wht we computed bove, we hve 2 ([, b, c] + [c, d, e]) = ( ([, b, c] + [c, d, e])) = ([b, c] [, c] + [, b] + [d, e] [c, e] + [d, e]) = [b, c] [, c] + [, b] + [d, e] [c, e] + [c, d] = (b c) ( c) + ( b) + (d e) (c e) + (c d) = 0. Tsk 12. Compute 2 (tht is, operte twice) on the following simplicil complexes. (1) σ = [, b] + [b, c], (2) σ = [, b, c] (use the work from Exmple 3.11), (3) σ = [, b, c, d] (use your work from Tsk 11). In fct, we lwys hve tht for every chin σ, we hve tht 2 σ 0. Tsk 13. Now tht you hve computed sufficient mount of boundries, try to use your intuition to determine the coefficients of the vertices in the boundry of the following 1-chins:

19 INTRODUCTION TO SIMPLICIAL COMPLEXES 19 b c d e g i f h j b d c f e g h SIMPLICIAL COLLAPSE Definition 3.14 (mximl element). Let K be simplicil complex. A fce of K is clled mximl element of K if it is not fce of ny simplex of K, except itself. The simplicil complex pictured below hs 5 mximl elements: the tetrhedron B, E, F, G, the tringle A, E, H nd the three segments B, C, C, D nd B, D. The tringle B, G, E is not mximl becuse it is fce of the tetrhedron. Definition 3.15 (free fce). If is mximl element of simplicil complex K, fce b of is clled free fce if b nd b is not contined in ny other simplex of K. In our exmple, the segment B, C hs no free fces, while the tringle A, E, H hs three free fces (its edges). Tsk 14. In the simplicil complex K pictured below, identify six different pirs (, b) where is mximl element of K nd b is free fce of. Question 3.16 (Chllenge Question). True or flse? If is free fce of mximl element b, then hs codimension 1. Tht is, the dimension is the simplex is 1 less thn the dimension of the simplex b. Definition 3.17 (simplicil collpse). Let K be simplicil complex. Suppose tht is mximl element of K nd b is free fce of. The ct of removing the fces {, b}, replcing K by the simplicil complex K {, b} is clled simplicil collpse.

20 20 CASEY KELLEHER AND ALESSANDRA PANTANO You cn think of simplicil collpse s removl of mximl element of K nd its free fce, by pushing in the free fce, until the entire mximl simplex disppers. Note: ll the remining fces of the mximl element remin. Severl exmples of simplicl collpses re shown in the pictures below. We iterte the construction until no more free fces re found (thus reducing the originl simplicil complex to its bone structure.) Tsk 15. Collpse the open book, until there re no free fces left.

21 INTRODUCTION TO SIMPLICIAL COMPLEXES 21

22 22 CASEY KELLEHER AND ALESSANDRA PANTANO