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Two-dimensional weak pseudomanifolds on eight vertices

BASUDEB DATTA and NANDINI NILAKANTAN

Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India E-mail: dattab@math.iisc.ernet.in; nandini@math.iisc.ernet.in

MS received 20 September 2001

Abstract. We explicitly determine all the two-dimensional weak pseudomanifolds on 8 vertices. We prove that there are (up to isomorphism) exactly 95 such weak pseudo- manifolds, 44 of which are combinatorial 2-manifolds. These 95 weak pseudomanifolds triangulate 16 topological spaces. As a consequence, we prove that there are exactly three 8-vertex two-dimensional orientable pseudomanifolds which allow degree three maps to the 4-vertex 2-sphere.

Keywords. Two-dimensional complexes; pseudomanifolds; degree of a map.

1. Introduction

Recall that a simplicial complex (in short, complex) is a collection of non-empty finite sets such that every non-empty subset of an element is also an element. Fori≥0, the elements of sizei+1 are called thei-simplices of the complex. Fori =1,2, thei-simplices are also called the edges and triangles of the complex, respectively. For a complexX, the maximum ofk such thatXhas ak-simplex is called the dimension ofX. The union of all the simplices of a complexXis called the vertex-set ofX and is denoted byV (X). Elements ofV (X)are called vertices ofX. A complexXis called finite ifV (X)is a finite set. Ak-simplex{v0, . . . , vk}of a complex is also denoted byv0· · ·vk.

IfX1andX2 are two complexes, then a simplicial map fromX1 toX2 is a mapϕ: V (X1)V (X2)such thatσX1impliesϕ(σ )X2. A bijectionπ:V (X1)V (X2) is called an isomorphism if bothπandπ1are simplicial. Two complexesX1, X2are called (simplicially) isomorphic when such an isomorphism exists. We identify two complexes if they are isomorphic. An isomorphism from a complexXto itself is called an automorphism of X. All the automorphisms of X form a group, which is denoted by Aut(X). Two simplicial mapsf, g:X1X2are said to be equivalent (denoted byf ∼=g) if there existϕ∈Aut(X1)andψ∈Aut(X2)such thatψfϕ =g.

Ad-dimensional simplicial complexXis called ad-dimensional weak pseudomanifold if each simplex ofXis contained in ad-simplex ofXand each(d−1)-simplex ofXis contained in exactly twod-simplices ofX. Ad-dimensional weak pseudomanifoldXis called a pseudomanifold (without boundary) if for any pairσ,λofd-simplices ofX, there exists a sequenceτ1, . . . , τnofd-simplices ofX, such thatσ =τ1,λ=τnandτiτi+1

is a(d−1)-simplex ofXfor 1≤in−1. (P1andP2, given in §2, are pseudomanifolds butQ1andQ2are not.)

A simplicial complex is usually thought of as a prescription for the construction of a topological space by pasting together geometric simplices (see §3 for finite complexes, 257

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[8] and [11] for the general case). The space thus obtained from a complexKis called the geometric carrier ofKand is denoted by|K|. We also say thatKtriangulates|K|.

For any setV ond+2 (≥2) elements, letKbe the simplicial complex whose simplices are all the non-empty proper subsets ofV. Then|K|is homeomorphic to the sphereSd. This complex is called the standardd-sphere and is denoted bySdd+2(V )or simply by Sd+d 2. A finite complexXis called a combinatoriald-sphere, if|X|is PL-homeomorphic to|Sd+d 2|[10]. Clearly, a finite one-dimensional complex is a combinatorial 1-sphere if and only if it is a pseudomanifold. Forn≥3, the combinatorial 1-sphere withnvertices is unique and is denoted byCn. The complexCnis also called ann-cycle. Ann-cycle with edgesv1v2, . . . , vn−1vn, vnv1is also denoted byCn(v1, . . . , vn).

Ifvis a vertex of a simplicial complexX, then the link ofvinX, denoted by LkX(v), is the complex whose simplices are those simplicesτ ofXsuch thatv 6∈ τ and{v} ∪τ is a simplex ofX. The number of vertices in the link ofvis called the degree ofvand is denoted by deg(v). Clearly, the link of a vertex in ad-dimensional weak pseudomanifold is a(d−1)-dimensional weak pseudomanifold.

A finite simplicial complex X is called a combinatorial d-manifold if |X| is a d- dimensional PL-manifold (without boundary), i.e., LkX(v)is a combinatorial(d−1)- sphere for each vertexv inX [7,10]. So,X is a combinatorial 2-manifold if the link of each vertex is a cycle. We also know (e.g., see [7]) that a finite simplicial complexKis a combinatorial 2-manifold if and only if|K|is a two-dimensional topological manifold.

A vertex of a finite two-dimensional weak pseudomanifold is called singular if its link is not a cycle (and hence consists of more than one cycle). So, a two-dimensional weak pseudomanifold is not a combinatorial manifold if and only if it contains a singular vertex.

(In each ofP1, P2, Q1andQ2, 7 is a singular vertex.)

A combinatorial 2-manifoldXis calledd-equivelar if each vertex ofXhas degreed. A combinatorial 2-manifold is called equivelar if it isd-equivelar for somed.

If the number ofi-simplices of ad-dimensional finite complexXisfi(X)(0≤id), then the numberχ(X):=Pd

i=0(−1)ifi(X)is called the Euler characteristic ofX. IfKis ad-dimensional oriented pseudomanifold, then [zK] generatesHd(K,Z), where zK:=P

σd∈Kσd(summation is taken over all the positively orientedd-simplices). Let KandLbe two orientedd-dimensional pseudomanifolds. Ifϕ:KLis a simplicial map thenϕd:Hd(K,Z)→Hd(L,Z)is a homomorphism and hence there existsm∈Z such thatϕd([zK])=m[zL]. Thismis called the degree ofϕand is denoted by deg(ϕ) [11].

LetKbe a two-dimensional pseudomanifold andϕ:KS42be a simplicial map. If ϕ(u)6=ϕ(v)for each edgeuvofKthenϕis called a 4-coloring [12].

It is known (e.g., see [3,5]) that if the number of vertices of a two-dimensional weak pseudomanifoldMis at most 6 thenMis a combinatorial 2-manifold andMis isomorphic toS1, . . . , S4orR1(given in §2).

In [5], we have seen that there are exactly nine 7-vertex combinatorial 2-manifolds and four 7-vertex two-dimensional weak pseudomanifolds which are not combinatorial 2-manifolds. Among the four non-manifolds two are pseudomanifolds, which triangulate the pinched sphere (the space obtained by identifying two points ofS2).

In [6], we have determined all the equivelar combinatorial 2-manifolds on at most 11 vertices. There are 27 such equivelar combinatorial 2-manifolds.

Altshuler and Steinberg [2] showed that there are fourteen 8-vertex combinatorial 2- spheres. Cervone [4] showed that there are exactly six 8-vertex combinatorial 2-manifolds, which triangulate the Klein bottle. It is known (e.g., see [7,9]) that there does not exist

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any 8-vertex combinatorial 2-manifold of Euler characteristic−1. Here, we classify all the two-dimensional weak pseudomanifolds on 8 vertices. More explicitly, we prove:

Theorem 1.1. There are exactly 44 distinct combinatorial 2-manifolds on 8 vertices, namely,S10, . . . , S23,R5, . . . , R20,T2, . . . , T8,K1, . . . , K6andD (given in §2). Theorem 1.2. There are exactly 51 distinct 8-vertex two-dimensional weak pseudomani- folds which are not combinatorial 2-manifolds, namely,P3, . . . , P39,Q3, . . . , Q16(given in §2).

COROLLARY 1.3

LetMbe a two-dimensional weak pseudomanifold onn (≤8)vertices.

(i) If|M|is a manifold then|M|is homeomorphic to the 2-sphere(S2), the real projective plane(RP2), the torus(S1×S1), the Klein bottle(K)or the space consisting of two disjoint 2-spheres.

(ii) If|M|is not a manifold then|M|is homeomorphic to the pinched sphere(P ),RP2#P, P#P,RP2#P#P,K#P,(S1×S1)#P, the union of twoS2’s having one, two, three or four points in common or the union ofS2andRP2having three points in common (given at the end of §2).(Here,A#Bdenotes the connected sum ofAandB). Ifϕ:KS42is a simplicial map, whereK is an oriented 8-vertex two-dimensional pseudomanifold, thenf2(K)≤18 and hence deg(ϕ)≤4. Here we prove:

Theorem 1.4. Letϕ:Kn2S42be a simplicial map, whereKn2 is a two-dimensional oriented pseudomanifold onnvertices. Letf,gandhbe as in Example 2.1. Ifn8 then deg(ϕ)≤3(and hence≥ −3). Equality is attained here if and only ifϕis equivalent to f,gorh.

Remark 1.5. Observe thatP3, . . . , P39 (in Theorem 1.2) are pseudomanifolds, whereas Q3, . . . , Q16 are not pseudomanifolds. Among the pseudomanifolds, P3, . . . , P20, P28, . . .,P36andP39are orientable andP21, . . . , P27,P37andP38are non-orientable.

Remark 1.6. IfMis an 8-vertex two-dimensional weak pseudomanifold, then it is easy to see thatχ(M)lies between−1 and 4. IfMis a combinatorial 2-manifold then, from Theorem 1.1,χ(M)is 0,1,2 or 4. However, by Theorem 1.2, there exist weak pseudo- manifolds with Euler characteristic−1, . . . ,3.

Remark 1.7. Theorem 1.1, Theorem 1.2 and Proposition 3.1 together with the results in [3], which determine all thed-dimensional weak pseudomanifolds on at mostd +4 vertices, classify all thed-dimensional (d 6= 3) weak pseudomanifolds on less than or equal to 8 vertices. Moreover, Theorem 1.1 together with Proposition 3.1, the results in [1], which classify all the combinatorial 3-manifolds on at most 8 vertices and the results in [3] classify all the combinatorial manifolds on less than or equal to 8 vertices.

In §2 we present all the two-dimensional weak pseudomanifolds on at most 8 vertices.

In §3 we give some definitions, constructions and results which we shall need later. In

§4 we consider combinatorial manifolds and prove Theorem 1.1. In §5 we consider weak pseudomanifolds which are not combinatorial manifolds and prove Theorem 1.2. In §6 we prove Corollary 1.3 and Theorem 1.4.

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2. Examples

In Theorems 1.1 and 1.2 we have stated that there are 95 two-dimensional weak pseudo- manifolds on 8 vertices. In this section we present all these 95 weak pseudomanifolds.

We also present all the 18 two-dimensional weak pseudomanifolds on less than or equal to 7 vertices. The degree sequences are presented parenthetically below the figures. For 0≤i≤7,iin the figures represents the vertexvi. At the end of this section, we present

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7

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2 3 1

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7

the geometric carriers of all the weak pseudomanifolds on 8 vertices. At the beginning we present three degree 3 maps (which we have mentioned in Theorem 1.4) to the 4-vertex 2-sphere.

Example 2.1. LetS1=S42({a, b, c, d})with the positively oriented 2-simplicesabc,acd, adb,bdc. LetS15,P34andP35be as given below:

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(a) Consider the orientation onS15given by the positively oriented 2-simplices 176, 160, 064, 104, 143, 132, 234, 524, 546, 562, 267, 127. Letf:S15S1be the simplicial map given byf (1)=f (5)=a,f (0)=f (2)=b,f (4)=f (7)=candf (3)= f (6)=d.

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0

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(b) Consider the orientation onP34given by the positively oriented 2-simplices 475, 467, 765, 056, 015, 061, 162, 426, 024, 043, 453, 135, 173, 037, 072, 127. Letg:P34S1

be the simplicial map given byg(0)=g(4)=a,g(1)=g(7)=b,g(2)=g(5)=c andg(3)=g(6)=d.

(c) Consider the orientation onP35given by the positively oriented 2-simplices 523, 537, 572, 274, 704, 760, 673, 163, 130, 203, 102, 124, 146, 564, 506, 540. Leth:P35S1

be the simplicial map given byh(1)=h(5)=a,h(2)=h(6)=b,h(3)=h(4)=c andh(0)=h(7)=d.

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Then deg(f )=deg(g)=deg(h)=3. The mapf is a 4-coloring butgandhare not.

3. Preliminaries

For a finite simplicial complexX, ifni (>0) is the number of vertices of degreedi and d1 > d2 >· · ·, thend1n1. . . . .dknk is called the degree sequence ofX, wherePk

i=1ni is equal to the number of vertices ofX.

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27 (7.6.5)6

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7

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Ifmi (>0) is the number of singular vertices of degreeci in a two-dimensional weak pseudomanifold X andc1> c2 >· · · then cm11. . . . .cmkk is called the singular degree sequence ofX, wherePk

i=1mi is equal to the number of singular vertices ofX.

IfX is a finite simplicial complex then one defines a geometric realization of X as follows: LetV (X)= {v1, . . . , vn}. We choose a set ofnpoints{x1, . . . , xn}inRN (for someN) in such a way that a subset S = {xj1. . . , xji+1}of i+1 points is affinely independent ifσ =vj1· · ·vji+1 is a simplex ofX. The convex set spanned bySis called the geometric carrier ofσ or the geometric simplex corresponding to σ and denoted

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2 3

0 5

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4

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3 6

1

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0

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7

1 7

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1

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5

4 5

0

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3

1 3

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0

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5 6

0

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1 2 3

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5

4 0

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3

1 2 3

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4 0

1 2 3 4

1 7

0

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1 2 3

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5 6

0 7

5

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1 3 2

4

6 5

7 0

1 2

Q13(6.5.3)3

3

1

2 1

4

6 5

0

4 7 2 4

7 1

2 3

5 6

7 0

Q10 (6.4 )2 6

7

3 0

0

6 2

7

1 1

4

2 6 7

4

7

5

P38 (7.6 )6 2

3

6 2 1

4 7 5

0 7

1 2 3

4 5

7 6

0 7

6 1

2 3

1 5

4 0

7 4

1 1

2 3

5 6

7 0

Q7 (7.4.3 )24 2 Q8(7.6.4.3)5 Q9 (6.5.4.3 )2 2 2 2 Q6 (7.4.3 )24 2

by|σ|. SinceX is finite we can chooseN so thatσγ = ∅implies|σ| ∩ |γ| = ∅. The set X := {|σ| : σX, σγ = ∅ ⇒ |σ| ∩ |γ| = ∅}is called a geometric simplicial complex corresponding toXor a geometric realization ofX. The topological space|X|:= ∪σ∈X|σ|is called a geometric carrier ofX. Clearly, if two finite complexes have a common geometric realization, then they are isomorphic and isomorphic finite complexes have homeomorphic geometric carriers [8]. We identify a complex with its geometric realization.

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S2 S2

P RP2

# P # P RP2 RP2

RP2

# P P # P

K # P # P K

S

S 2

2

S x S1 1

( )

RP2 S2

1 0

5 7 6

4 5

7

6 2

1 2

3 0 1

2 3

7

6 5

4 0

3 1 2 7

3 2 5 1 0

0 7

6 4

5 1

Q

14(7.5 )44

Q

15(6 )8

Q

16(7.6.3)3 4

S2

S2

S2

S2

IfMis ann-vertex two-dimensional weak pseudomanifold withf1edges andf2trian- gles then 3f2=2f1. Therefore,

χ(M)=nf1+f2=nf1

3 =nf2

2 . (1)

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In [5], we have seen the following:

PROPOSITION 3.1

There are exactly 13 distinct two-dimensional weak pseudomanifolds on 7 vertices, namely, T1,R2,R3,R4,S5, . . . , S9,P1,P2,Q1andQ2.

Altshuler and Steinberg [2] showed the following:

PROPOSITION 3.2

There are exactly 14 distinct combinatorial 2-spheres on 8 vertices, namely,S10, . . . , S23. IfX1, X2are two simplicial complexes with disjoint vertex sets, then their joinX1X2

is the complex whose simplices are those ofX1andX2and the unions of simplices of X1with simplices ofX2. If bothX1andX2are pseudomanifolds then so isX1X2[3].

Observe thatS20({a, b})S20({c, d})=C4(a, c, b, d).

IfMis a two-dimensional simplicial complex andτ=abcis a 2-simplex then(M\τ)(∂τv)denotes the two-dimensional complex whose 2-simplices areabv,acv,bcvand those ofMother thanτ, wherev 6∈V (M). This complex is said to be obtained fromM by starring the vertexvinτ. Observe thatR2is obtained fromR1by starring the vertex 0 in 123. Similarly, if abis an edge (contained in abcandabd) andv /V (M)then (M!\{ab})∪(C4(a, c, b, d)∗v)denotes the two-dimensional complex whose 2-simplices areacv,adv,bcv,bdvand those ofMother thanabcandabd. This complex is said to be obtained fromMby starring the vertexvin the edgeab. The complexR3is obtained fromR1by starring the vertex 0 in the edge 12.

LetM,Nbe two simplicial complexes withσ1, . . . , σmMandτ1, . . . , τmN. We say(M, σ1, . . . , σm)and(N, τ1, . . . , τm)are isomorphic (denoted by (M, σ1, . . . , σm)

∼=(N, τ1, . . . , τm))if there exists an isomorphismϕfromMtoN such thatϕ(σi)=τi for 1≤im.

LetMbe ad-dimensional weak pseudomanifold. Letuandvbe two distinct vertices ofMsuch thatuvis not an edge. IfV (Lk(u))V (Lk(v))= ∅then define the complex Me= {τ ∈M:u6∈τ, v6∈τ} ∪ {(τ\{u})∪ {w}:uτ} ∪ {(τ\{v})∪ {w}:vτ}. ThisMe is called the simplicial complex obtained fromMby identifyinguandv(to a new vertex w). Observe thatP1is obtained fromS20by identifying vertices 0 and 3 ofS20andP2is is obtained fromS16by identifying vertices 0 and 5 ofS16.

Ifτ1=abcandτ2=xyzare two disjoint 2-simplices of a two-dimensional simplicial complexMandv 6∈V (M)then(M\{τ1, τ2})∪((∂τ1∂τ2)v)denotes the complex whose 2-simplices areabv,acv,bcv,xyv,yzv,xzvand those ofMother thanτ1andτ2. Observe that we getP33fromP2andP39fromT1by this process.

From these definitions one gets the following:

PROPOSITION 3.3

LetMebe obtained fromMby identifying two verticesuandv. (a) IfMis a(weak)pseudomanifold, then so isMe.

(b) If Neis obtained fromN by identifying two vertices u1and v1 and(N,{u1},{v1})

∼=(M,{u},{v}), thenNe∼=Me.

(c) IfM is a two-dimensional pseudomanifold and bothuand vare non-singular then

|M|e is homeomorphic to the connected sum of|M|and the pinched sphere.

(13)

PROPOSITION 3.4

LetMbe a two-dimensional simplicial complex.

(a) Letτ be ani-simplex(1≤i≤2)andMebe obtained fromMby starring a vertex in τ. IfMis a weak pseudomanifold, pseudomanifold or combinatorial 2-manifold then so isMewith the same geometric carrier.

(b) Letτ12are disjoint 2-simplices ofMandu, v, w6∈V (M). LetMb:=(M\{τ1, τ2})∪

((∂τ1∂τ2)w). LetNbe the complex obtained fromMby starringuinτ1andvin τ2. LetNebe obtained fromNby identifyinguandv. IfMis a(weak)pseudomanifold then so isNeandNe∼=M (b and hence|M|b is homeomorphic to the connected sum of

|M|and the pinched sphere wheneverMis a pseudomanifold).

PROPOSITION 3.5

LetM1andM2be two-dimensional weak pseudomanifolds.

(a) LetMej be obtained fromMj by starring a vertex on ani-simplex(1≤i≤2) σj for j =1,2. If(M1, σ1)∼=(M2, σ2), thenMe1∼=Me2.

(b) If (M1, σ1, τ1) ∼= (M2, σ2, τ2) and u1, u2/ V (M1)V (M2), where σj, τj are disjoint 2-simplices ofMj forj =1,2, then(M1\{σ1, τ1})∪((∂σ1∂τ1)u1)∼= (M2\{σ2, τ2})∪((∂σ2∂τ2)u2).

PROPOSITION 3.6

Letϕ:KLbe a simplicial map of degreed >0, whereKandLare two-dimensional oriented pseudomanifolds. For a vertexvofL, letSv := {σ ∈ K : ϕ(σ )is a 2-simplex containingv}.

(a) Ifσ is a 2-simplex ofLthenϕ1(σ )contains at leastdsimplices.

(b) If for some 2-simplexσofL,ϕ1(σ)containsdord+1 2-simplices, sayσ1, . . . , σd

(orσd+1), thenσi andσj have at most one vertex in common fori6=j.

(c) If for some vertexvof degreecofL,Svcontains the 2-simplicesτi∪ {ui}i ∪ {vi}, whereϕ(ui)=ϕ(vi)=vfor 1ip, then #(Sv)cd+2p.

Proof. (a) follows from the definition of the degree of a simplicial map.

If possible letϕ(uvx)=ϕ(uvy) =σ. Ifϕ2:C2(K)C2(L)is the homomorphism induced byϕthen, for any orientations ofKandL,ϕ2(+uvx)= −ϕ2(+uvy)inC2(L). So, ifmis the coefficient of+σinϕ2(C2(K))then|m| ≤d+1−2 and hence deg(ϕ)d−1, a contradiction. This proves (b).

By the same argument as in (b),ϕ2(+τi ∪ {ui})= −ϕ2(+τi∪ {vi}), for 1≤ ip. Therefore, for each 2-simplexσ containingv (as the degree ofϕ isd) #21(σ)\{τi∪ {ui}, τi∪ {vi}: 1≤ip})d. This proves (c). 2

For a simplicial mapϕ: KL, ad-simplexσis said to be collapsing ifϕ(σ)is not a d-simplex. Letϕ:KS42be a simplicial map, whereKis anm-vertex two-dimensional weak pseudomanifold andS42is the standard 2-sphere (with 2-simplicesσ1, . . . , σ4). The mapϕ is said to be of type (n1, n2, n3, n4)ifni is the number of triangles (inK) with imageσi (1≤i≤4) andn1n2n3n4.

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