NEKHBORLY COMBINATORIAL 3-MANIFOLDS

WITH 9 VERTICES

For etc=ry complex C‘ and every simp?ex A cz C, we define star f A, I”)

to be the union of those simpiices in C‘ that contain A as ti Gt‘e; ~1 star

CA. C-3 is defined to be the :~rnallest complex th;tt contains star (A, 0

{i.e.. the members of cl star (A, C’) are thasc of st:lr (A, (1 and their

- --. .- _ . . . - ._ .__ .- .L-- -. - ___- _ -_. _ . . _* _ _ _. _ . - _ --. -- I -, -_ _.._. ._ . _ _ _

Vt!ftlwS i

5 6 7 8 9 ._ _ _. _. . -,.. -.

3. Directly obt;ainrrble manifolds

3. Proof of Theorem I. 1.

P- M u-l r.

t

Z =

x Y i l l

E x T

x p-

133

The 50 dirt:&. obtain~~ble S9’s previously obtained are easilv idczl-

tified in the new li*.t by the determinant of the edge-valence m&ix. The

remaining 3-mmifc id A’& (rice Table 2). has the highest determinant ot

the S l ~stscs. A priori, all that we know about A’& is th;nt it is not ti di-

rectly obtainable sphere. and therefore it may be either a sphere th;it is

not direstly obtamablc, or a manifold that is not a sphere. However, it

is e,tsy t13 &tlck that S& is not orientable, and hence it is not a sphere.

The 3-:si:nplices .W 2’, 0. Q, S, 7’. C’ and Y of A’& Llrm a solid Klein

bottle hf. The boundary complex of K is a 2dimensional Klein btittle wi rh 8 vt’rtrces. 23 cadges and 16 triangles. and is ‘;hcreiore a minimal

triLngulation of ;1 topological Klein bottle (see [9] ). The proof oi Theorem 1_ 1. and therefore also of Theorem 1.2, is thus

Comple tc.

I l I E;:ih 2-sphrrc Yi in Fig. 1 15 accomplznied by the 8-tuple of the

valences of the vertices m c), Q the 8-tuple that corresponds to 4i ( 1 < i

5 7 ) has bern &noted by C; in [4] ). For each A9 and each vertex _

u f A9, the H-tuple that corresponds to link (u, A9 ) is direAy readable: from the edge-valence matrix 0fQ in the tnarw~r explliined in [4] .

?ioticc that t’7 md qq s!we the same 8-tupk. this accounting ftir the

ftic‘t that the ,i-q~h~r~ .V& qpe;!re d in f4 J (where it is denoted by A’)

although it is not polytoptil. L

(2) ‘Lb/e 3 summarizes the types of links c >rrespol‘ding to the v&-

tic’ej of AT ( 24 5 i < S 1). and is thus a continuation of [ 4. Table 31 ’

where d similar summary for the polytopal spherl:s A’:. . . . . .V;, was

given. From thlo summary, we see that no 1V has a vertex u such that link iu, X9 I is isarrjorphic to e’ 13 , i.e., to the bountlary complex of a 3-

dimension;ri bipyrrimid with 8 vertices. From [ loj and 1 I?], we see that

nt*ighborly 3-manifolds with seven or eight vertices possess a similar :Jro-

pcrty . This leads us to venture the following conjecture:

.vI” 1 _ __

241

3 1. b.!

26

1 2 ,I

28

?Q

3?

33

32

33

34

35

36

37

3%

39

40

4i

41

II

44

4

46

P (1 e? J . “3 “4 4G ‘4 ‘F7 . % ‘9 % ql “12 ‘13 q4

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7 IV 6 1

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1 E 4 1 1 f

? _y4 3 I

4 5 L I 2

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4 : 1 2 2 ..-- . ._.._ I _. .__.._.__.__. ._._.__. 1 ..-..__- . . _ __ - .______._ -. .__. .-- ., __.___ _-.. _ .- . . . _ __ - _

5 m 2 2 2 1 .- __- - .- --.- .--- -.- .I A ._._._ _ --_ -___- ._-_ __-- ._.__ __ _._ _.._^.. _ ‘.__ _. __ ._.__ ^ _. . ,.

1 1 1 Z B i I 1 __ .^ _. _“_. . “_^.. ___._ _ _.._ l,-.-...._* _ _ . - _~

2 1 3 2 1

1 s 2 1 ._ .__._ -.- ..__ -_ _ --_- ._____ ._..___ .- .-__ _ ___. _,__-._I ..,._.- _ . ..- .._.. _ __.. ._ .

9 2

-7,

L 0. 2 1

1 2 1 1 2 I 1

2 I 1 : 1 3 I .-. __ ._._ ___. --_--_ _ __.I ,I ..._-__y __ __ _... ___..._ _ ._-..._. ___“__. .-. ____ -_ _._e..* - .-. _-... . . I -

2 2 2 I I 1 ---- _--___- __.-_--..----_-.~-__._--_-._-__-___II . . . . . . .I. _-^* ._.._ _. ._...I -.-_--.....nI__..-I.

- -. .-.“..a-.-. --..

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2 1 1 2 3

q I ‘1 P, “f t4 I 9 ._e_ -t.. _.- _ . . . . .._I . _ __ __

43 1 2 . . -_._..

4s $ I 2 ._.\ - _

49 2 . . . .

50 I_ - ~.. __

51 - . . - _ _ _ . .

I . .

“6 c7 L’s, :* , .* 9 % w “i2 . 41 1 (’ ;t 4

_ .-. . . - *

2 c 7

r - _ _- - - -. . -_ ,_. --_- ,~..._ _

2 ? I . _.. _

I 4 _ _ .- _ .

3 _ . _ _ _ _ ^ . _ ._

9 _ . _ .I_.. _.-