Homologies of moduli space $\mathcal{M}_{2,1}$

0.19 MB·English

by  Yury Kochetkov

HOMOLOGIES OF MODULI SPACE M2,1
YURYKOCHETKOV
3
1
0
2

n Abstract. WeconsiderthespaceM2,1—theopenmodulispaceofcomplex
a curves of genus 2 with one marked point. Using language of chord diagrams
J we describe the cell structure of M2,1 and cell adjacency. This allows us to
5 constructmatricesofboundaryoperatorsandcomputeBettynumbersofM2,1
2 overQ.

]
O 1. Introduction
C A graph G is correctly embedded into a compact topological oriented surface
. S, if its complement is homeomorphic to a 2-dimensional disk. If a graph G is
h
t correctly imbedded into S and lengths of its edges are given, then the Jenkins-
a
Schtrebel constructionallows one to uniquely define a complex structure on S (see
m
[2]). In what follows we will assume that the sum of lengths of edges is 1.
[
Let M2,1 be the open moduli space of genus 2 complex curves with one marked
1 point. A topological cell complex Mc2o,1mb, which is homeomorphic to the space
v M2,1,hasthefollowingcombinatorialdescription(see[1],Chapter4,forexample).
9
Let us consider the set of all pairwise nonisomorphic graphs, correctly embedded
5
0 into S2 — topological compact genus 2 surface. Isomorphism here is the isomor-
6 phism of embedded graphs, i.e. isomorphism must preserve the cyclical order of
. edges under the counterclockwise going around of each vertex.
1
0 ToagraphΓwithsedges,correctlyimbeddedintoS2,wecorrespondthesimplex
3 ∆Γ, which is isometric to standard simplex
:1 {x1,...,xs ∈Rs|x1+···+xs =1, x1,...,xs >0}.
v
i Here numbers x1,...,xs are lengths of edges of Γ. Cells of the highest dimension
X 8 correspond to correctly imbedded 3-valent graphs (a graph is called 3-valent, if
r each its vertex has degree 3). A 3-valent graph correctly imbedded in to S2 has 6
a
vertices and 9 edges.
If a correctlyimbedded graphΓ′ is obtained by a contractionof some edge ofΓ,
then to Γ′ some face of ∆Γ is corresponded.
A correctly imbedded graph Γ may have a nontrivial group of automorphisms.
We can define an action of this group on the simplex ∆Γ also. The cell of the
spaceMc2o,1mb iseitherthe simplex∆Γ,ifthe correspondinggraphdoesn’thaveany
nontrivial automorphisms, or the factor of ∆Γ by the action of the group.
ThusdefinedspaceMcombisnoncompact. Inthisworkwewillfinditshomologies
2,1
over Q.
2. Graphs, glueings and diagrams
The topological surface S2 with correctly imbedded 3-valent graph can be re-
alized as a glueing of 18-gon. Here the embedding correctness is automatically
1
2 YURYKOCHETKOV
fulfilledandthe 3-valencyandthe genus2 conditiongive9pairwisenonisomorphic
glueings (two glueings of 2n-gon are isomorphic, if some rotation of 2n-gon trans-
forms one onto another). The groupof automorphisms of a graphis a cyclic group
of automorphisms of the corresponding diagram of glueing. Combinatorics of cells
of the highest dimension is described in [3].
A contraction of an edge of graph corresponds to a contraction of the pair of
identified sides of corresponding glueing. Thus cells of dimension 7 correspond to
glueings of 16-gons and cells of dimension 6 — to glueings of 14-gons.
We will use schemas of chord diagrams for enumeration of glueings.
Example 1. Gaussian word a1a2a3a−11a4a−21a−41a−31 that describes the glueing of
octagon
(cid:27)a3
a4 (cid:0) @ a1
(cid:0) @
1
(cid:0)(cid:9) @R
we will write Gaussian q
a2 a2 word as string 12314243,
and represent the glueing
? ?
as a chord diagram:
@I (cid:0)
a@4 (cid:0)a3
@ -(cid:0)(cid:9)
a1
simplifying it to a scheme
A (cid:1)q
A(cid:1)
(cid:1)A
(cid:1) A
The point at diagram denotes the beginning of clockwise numeration of chords.
All glueings we will work with are enumerated in Appendices 1 – 6. A glueing
will be denoted γ(k) , where k is the number of edges of corresponding graph and
l
l is the number of this glueing in the list of glueings of 2k-gons. For each glueing
we give its chord diagram, Gaussian word and group of automorphisms (if it is
nontrivial). Gaussian word defines the numeration of chords, which will be called
standard.
All chord schemas, enumerated in Appendices, define a genus 2 curves with
correctly embedded graphs. Each diagram γ(k) , k = 8,7,6,5,4 is obtained from
i
some scheme γ(k+1) by deletion of a chord.
j
A scheme γ(k) defines a (k −1)-dimensional cell in the space Mcomb. Cells
i 2,1
that correspond to genus 2 schemas, obtained from γ(k) by deletion of a chord,
i
constitute the (k−2)-dimensional boundary of our cell.
A glueing of 2n-gon (a chord diagram of glueing) will be called symmetric if it
has a nontrivial groupof automorphisms. This groupis a cyclic groupof rotations
of 2n-gon. As sides of 2n-gon (chords of diagram) are numerated, then generator
of group of automorphisms defines a permutation from S . We will call a glueing
n
(scheme) even-symmetric, if this permutation is even, and odd-symmetric in the
opposite case.
HOMOLOGIES OF MODULI SPACE M2,1 3
A cell of the space Mcomb will be called simple, if this cell is a simplex. A
2,1
cell=factorized simplex will be called special even, if the corresponding glueing is
even symmetric, and special odd, if the corresponding glueing is odd symmetric.
The topological space Mcomb consists of (see Appendices 1–6):
2,1
• 9 cells ofdimension 8: 3 are simple, 5 are simplices, factorizedby action of
the group Z2, and one special cell is a simplex factorized by action of the
group Z3 (here all special cells are even);
• 29 cells of dimension 7: 24 cells are simple, 4 are simplices, factorized by
actionofthegroupZ2(thesecellsareeven),andonespecialcellisasimplex
factorized by action of the group Z4 (this cell — δ(7)20 is odd);
• 52cellsofdimension6: 41cellsaresimple,11arespecialcells—simplices,
factorized by action of the group Z2 (two of them — δ(6)39 and δ(6)40 are
even, all others are odd);
• 45 cells of dimension 5: 37 cells are simple, 5 are simplices factorized by
actionofthegroupZ2,oneisasimplexfactorizedbyactionofthegroupZ3,
one is a simplex factorized by action of the group Z4 and one is a simplex
factorized by action of the groupZ6 (two special cells — δ(5)39 and δ(5)41
are even, all others are odd);
• 21 cells of dimension 4: 14 cells are simple, 5 are simplices factorized by
action of the group Z2, one is a simplex factorized by action of the group
Z5 and one is a simplex factorized by action of the group Z10 (one special
cell — δ(4)19 is odd, all others are even);
• 4 cellsofdimension3: twoaresimple, oneisasimplex factorizedbyaction
of the group Z2 and one is a simplex factorized by action of the group Z8
(the cell δ(4)1 is special even and the cell δ(4)4 is special odd).
EulercharacteristicofopenmodulispacesM2,1 is9-29+52-45+21-4=4. Andits
”orbifoldic” Euler characteristic is
5 1 4 1 11 5 1 1 1
3+ + − 24+ + + 41+ − 37+ + + + +
(cid:18) 2 3(cid:19) (cid:18) 2 4(cid:19) (cid:18) 2 (cid:19) (cid:18) 2 3 4 6(cid:19)
5 1 1 1 1 1
+ 14+ + + − 2+ + = .
(cid:18) 2 5 10(cid:19) (cid:18) 2 8(cid:19) 120
This result is in agreement with Harer-Zagierformula [4].
3. Homologies
At first we’ll explain the notion of induced numeration.
Definition 1. The scheme,obtainedbydeletionofthei-thchordfromaschemeγ
will be denoted γ[i]. We can define a numeration of chords of γ[i] in the following
way: if a chord c has number j < i in γ, then c has the same number in γ[i]; if
a chord c has number j > i in γ, then c has number j −1 in γ[i]. Thus defined
numerationofchordsoftheschemeγ[i]willbecalledinduced. Wewillusenotations
γ[i]ind and γ[i]st (i.e. standard) when necessary.
If a cell δ corresponds to a scheme γ, then by δ[i] we will denote the cell that
corresponds to scheme γ[i].
Remark 1. As γ[i] = γ[p] for some p and q, then scheme γ[i] has the standard
q
numeration also. The standard and induced numerations are usually different.
4 YURYKOCHETKOV
Definition 2. Let C =C (Mcomb,Q) be the space of k-dimensional chains with
k k 2,1
rational coefficients. The boundary operator ∂k : Ck → Ck−1 maps each cell
k+1
δ(k)i into chain j=1αjδ(k)i[j] ∈ Ck−1. Coefficients αj are defined by following
conditions. P
(1) Let cells δ(k) and δ(k) [j] be simple. The identification of schemas
i i
γ(k +1)i[j]ind and γ(k +1)i[j]st defines some renumeration of chords of
the scheme γ(k+1)i[j]ind, i.e. defines a permutation σ ∈ Sk of parity p,
p=0,1. Set α :=(−1)j+p−1.
j
(2) Let the sell δ(k) be simple and the cell δ(k) [j] be special even — a sim-
i i
plex factorized by action of cyclic group of order m. The identification of
schemas γ(k +1)i[j]st and γ(k +1)i[j]ind doesn’t define the permutation
σ ∈ S uniquely, but all such permutation have the same parity p. Set
k
α :=(−1)j+p−1m.
j
(3) Letthesellδ(k) besimpleandthecellδ(k) [j]bespecialodd. Setα :=0.
i i j
(4) Let the cell δ(k) be special even: a cyclic group Z acts on the scheme
i r
γ(k+1) (and on the simplex ∆(k) ). The j-th chord either belongs to an
i i
orbit of cardinality r, or itself is an orbit. In the first case the deletion of
this chordgivesusanonsymmetricschemeandthe coefficientα isdefined
j
accordingtotherule(1)above. InthesecondcaseacyclicgroupZ ,where
q
r divides q, acts on the scheme γ(k + 1) [j] (and on the corresponding
i
simplex). If this scheme is even symmetric, then α := (−1)n+p−1q/r
j
(where parity is computed in above defined way). If this scheme is odd
symmetric, then α :=0.
j
(5) Let the cell δ(k) be special odd and a cyclic group Z acts on the scheme
i r
γ(k+1) (and on the simplex ∆(k) ). If the j-th chordbelongs to an orbit
i i
ofcardinality>1,then setα :=0. Ifthis chorditselfis anorbit,thenthe
j
schemeγ(k+1) [j]isoddsymmetricandacyclicgroupZ ,wherer divides
i q
q, acts on it. Now set α := (−1)j+p−1q/r (where parity is computed in
j
above defined way).
Remark 2. The union of special odd cells constitute a subcomplex in Mcomb.
2,1
Matrices of boundary maps are presented in Appendices.
Theorem 1. ∂k−1◦∂k =0.
Proof. Let γ be a (k+1)-scheme. Let us consider chords with numbers i and j.
We need to prove that the scheme γ[i][j] has zero coefficient in the sum ∂2(γ). If
schemas γ, γ[i], γ[j] and γ[i][j] are simple, then this statement is a consequence of
the analogous result for simplicial homologies. It means (if we ignore conditions
2-5)thatthesignofthepassageγ →γ[i]→γ[i][j]isoppositetothesignofpassage
γ → γ[j] → γ[j][i]. Thus, it remains to take into account symmetry conditions.
Let us consider two typical cases.
Schemas γ, γ[i] and γ[i][j] are simple and scheme γ[j] is odd symmetric. Let ϕ be
a generator of symmetry group of the scheme γ[j] and i = i1,...,im be the orbit
of i-th chord (here m is even and i2 =ϕ(i1), i3 =ϕ2(i1),...). Symmetry ϕ defines
a new numeration on γ[j] such, that i1-th chord now has number i2. New induced
numerationon γ[i][j] after deletion of i2-th (former i1-th) chord is the same as old
induced numeration on γ[j][i2]. It remains to note that in both cases we multiply
by (−1)i2−1 and that permutation defined by ϕ is odd.
HOMOLOGIES OF MODULI SPACE M2,1 5
Schemas γ and γ[i] are simple, scheme γ[j] is odd symmetric and scheme γ[i][j]
is even symmetric. As above, let ϕ be a generator of symmetry group of the
scheme γ[j] and i = i1,...,im (m is even) be the orbit of i-th chord. Schemas
γ[i1][j],...,γ[im][j]allaresomeschemeγ(k)s. Letpk betheparityofrenumeration
from γ[ik][j]st to γ[ik][j]ind. Now it is clear that m/2 of these pk are zeroes and
m/2 are units. (cid:3)
Theorem 2 is the main result of the work.
Theorem 2. Let M be the matrix of the boundary map ∂ , then
i i
rk(M8)=8, rk(M7)=20, rk(M6)=27, rk(M5)=17, rk(M4)=3.
Thus, Betty numbers are:
b8 =1, b7 =1, b6 =5, b5 =1, b4 =1, b3 =1.
4. Appendix 1. Schemas of nine-edge glueings.
γ(9)1 :................HA(cid:8)(cid:1)..A(cid:1)....HA(cid:8)(cid:1)....A(cid:1)....H(cid:8)....................(cid:8)Hr......A(cid:8)H(cid:1)....A(cid:1)..A(cid:8)(cid:1)H....A(cid:1)................ γ(9)2 :...................HL(cid:8)(cid:12)..L(cid:12)....L(cid:12)H(cid:8)..L(cid:12)..LL(cid:12)(cid:12)....H(cid:8)............(cid:8)Hr........L(cid:8)H(cid:12)..L(cid:12)..L(cid:12)..(cid:8)H..L(cid:12)..LL(cid:12)(cid:12)................. γ(9)3 :...................(cid:10)J(cid:8)H....(cid:10)J..(cid:8)H..(cid:10)(cid:10)JJ......(cid:8)H......(cid:8)H..(cid:18)(cid:19)..r..L(cid:12)(cid:8)H....L(cid:12)..L(cid:12)..(cid:8)HL(cid:12)....LL(cid:12)(cid:12).................
123142345678697895 Z2 123145235678649789 Z2 123145623678497589
γ(9)4 :.......................A(cid:1)(cid:17)Q..A(cid:1)....A(cid:1)(cid:17)Q....A(cid:1)..(cid:17)Q........(cid:17)Q(cid:8)H......B(cid:17)rQ(cid:8)H(cid:2)..B(cid:2)..B(cid:2)..(cid:17)QB(cid:8)H(cid:2)..B(cid:2)..BB(cid:2)(cid:2)....................... γ(9)5 :...................(cid:8)H@(cid:0)......(cid:8)H@(cid:0)(cid:28)\......(cid:28)\..(cid:8)H....(cid:28)\..(cid:8)H..(cid:28)\....(cid:0)@r(cid:8)H..(cid:28)\......(cid:0)@(cid:8)H....................... γ(9)6 :.......................P(cid:3)(cid:8)aA..(cid:3)..(cid:3)A(cid:3)..P(cid:3)(cid:8)aA..(cid:3)..(cid:8)A....P(cid:8)Aa..A(cid:8)....P(cid:8)raP....P(cid:8)..(cid:8)aP....(cid:8)A....(cid:8)aPA..(cid:8)A....(cid:3)aPA..(cid:3)..(cid:8)aP(cid:3)A(cid:3)..(cid:3)A..(cid:3).....................
123145672378495869 123145647812895697 Z2 123415624785973689 Z2
γ(9)7 :.......................Q(cid:24)Ar(cid:17)(cid:1)X....A(cid:1)..Q(cid:24)A(cid:17)(cid:1)X....A(cid:1)....Q(cid:24)(cid:17)X......Q(cid:24)(cid:1)A(cid:17)X....(cid:1)A..Q(cid:1)A(cid:17)....(cid:1)A....Q(cid:1)A(cid:17)..(cid:1)A......................... γ(9)8 :.......................(cid:16)CH!(cid:1)r..C..C(cid:1)C..(cid:16)CH!(cid:1)..C..H(cid:1)....(cid:16)H(cid:1)!..(cid:1)H....(cid:16)H!(cid:16)....(cid:16)H..H!(cid:16)....H(cid:1)....H!(cid:16)(cid:1)..H(cid:1)....C!(cid:16)(cid:1)..C..!H(cid:16)(cid:1)CC..(cid:1)C..C..................... γ(9)9 : ......(cid:10)......(cid:10)........(cid:10)......(cid:10)(cid:10)..................rJJ....J......J......J..........
123415672485968379 123415673859682479 Z2 123451672586937849 Z3
5. Appendix 2. Schemas of eight-edge glueings.
γ(8)1 :................B(cid:2)..B(cid:2)..B(cid:2)..B(cid:2)......(cid:21)(cid:20)..............(cid:10)Jr....(cid:10)J......(cid:10)J(cid:10)J................. γ(8)2 :................(cid:2)B..(cid:2)B..(cid:2)B..(cid:2)B......(cid:20)(cid:21)..............(cid:8)Hr........(cid:8)H................... γ(8)3 :................B(cid:2)..B(cid:2)..B(cid:2)..B(cid:2)......(cid:21)(cid:20)........(cid:23)(cid:22)r....(cid:2)B..(cid:2)B..(cid:2)B..(cid:2)B................
1231234567586784 1213234567586784 1231423456758678 Z2
6 YURYKOCHETKOV
γ(8)4 :................B(cid:2)..B(cid:2)..B(cid:2)..B(cid:2)......(cid:21)(cid:20)........(cid:23)....B(cid:2)r..B(cid:2)..B(cid:2)..B(cid:2)................ γ(8)5 :................(cid:3)H(cid:8)C..(cid:3)C..(cid:3)C(cid:3)C..H(cid:8)..............(cid:8)Hr......(cid:8)H.................... γ(8)6 :................(cid:3)H(cid:8)C..(cid:3)C..(cid:3)C(cid:3)C..H(cid:8)..............Hr......HC(cid:3)..C(cid:3)..C(cid:3)..C(cid:3)..............
1234124567538678 1213424567538678 1231423567548678
γ(8)7 :...................(cid:8)(cid:10)JH....(cid:10)J..(cid:8)H..(cid:10)(cid:10)JJ......(cid:8)H......(cid:16)(cid:8)H......Lr(cid:16)(cid:12)(cid:8)H....L(cid:12)..L(cid:12)..(cid:16)(cid:8)HL(cid:12)....LL(cid:12)(cid:12)................. γ(8)8 :..................."(cid:10)Jb....(cid:10)J.."b..(cid:10)(cid:10)JJ......"b......"(cid:8)Hrb......"(cid:8)Hb....................... γ(8)9 :...................(cid:8)(cid:10)JH....(cid:10)J..(cid:8)H..(cid:10)(cid:10)JJ......(cid:8)H......Pr(cid:8)H......LP(cid:12)(cid:8)H....L(cid:12)..L(cid:12)..P(cid:8)HL(cid:12)....LL(cid:12)(cid:12).................
1234512567386478 1213452568386478 1231452367486578
γ(8)10:...................(cid:8)(cid:10)H....(cid:10)..(cid:8)H..(cid:10)(cid:10)......(cid:8)H......(cid:16)Pr(cid:8)H......L(cid:16)P(cid:12)(cid:8)H....L(cid:12)..L(cid:12)..(cid:16)P(cid:8)HL(cid:12)....LL(cid:12)(cid:12)................. γ(8)11:..................."(cid:3)Cb..(cid:3)C..(cid:3)C(cid:3)C.."b........"b(cid:16)Pr......"bL(cid:16)P(cid:12)..L(cid:12)....L(cid:12)"b(cid:16)P..L(cid:12)..LL(cid:12)(cid:12)................... γ(8)12:....................J(cid:8)H....J..(cid:8)H..JJ......(cid:8)H......(cid:16)P(cid:8)H......rL(cid:16)P(cid:12)(cid:8)H....L(cid:12)..L(cid:12)..(cid:16)P(cid:8)HL(cid:12)....LL(cid:12)(cid:12)..................
1231456236748578 1231456236748758 1231456236784758
γ(8)13:......................QA(cid:17)(cid:1)....A(cid:1)..QA(cid:17)(cid:1)....A(cid:1)....Q(cid:17)......Q(cid:16)(cid:17)......BrQ(cid:16)(cid:2)(cid:17)..B(cid:2)..B(cid:2)....BQ(cid:16)(cid:2)(cid:17)..B(cid:2)..BB(cid:2)(cid:2).................... γ(8)14:......................#(cid:1)cA....(cid:1)A..#(cid:1)Ac....(cid:1)A....#c......r#c(cid:8)H......#c(cid:8)H........................... γ(8)15:......................QA(cid:17)(cid:1)....A(cid:1)..AQ(cid:17)(cid:1)....A(cid:1)....Q(cid:17)......QPr(cid:17)......QBP(cid:2)(cid:17)..B(cid:2)..B(cid:2)....QBP(cid:2)(cid:17)..B(cid:2)..B(cid:2)B(cid:2)....................
1234561267384768 1213456267384758 1231456237384768
γ(8)16:......................Q(cid:1)(cid:17)....(cid:1)..Q(cid:1)(cid:17)....(cid:1)....Q(cid:17)......Q(cid:16)rP(cid:17)......BQ(cid:16)P(cid:2)(cid:17)..B(cid:2)..B(cid:2)....QB(cid:16)P(cid:2)(cid:17)..B(cid:2)..BB(cid:2)(cid:2).................... γ(8)17:......................QA(cid:17)....A..AQ(cid:17)....A....Q(cid:17)......Q(cid:16)rP(cid:17)......BQ(cid:16)P(cid:2)(cid:17)..B(cid:2)..B(cid:2)....QB(cid:16)P(cid:2)(cid:17)..B(cid:2)..BB(cid:2)(cid:2).................... γ(8)18:................#@(cid:0)a......#S(cid:0)a......S..#(cid:3)a(cid:3)..(cid:3)S..(cid:3)..#(cid:0)a..S....#(cid:0)@ra....................
1231456723748568 1231456723784586 1213453672784586 Z2
γ(8)19:................!@(cid:0)c......!(cid:19)@c......(cid:19)..!CcC..C(cid:19)..C..!@rc..(cid:19)....!@(cid:0)c.................... γ(8)20:..................."@(cid:0)b......"T(cid:20)@(cid:0)b....T(cid:20)...."bT(cid:20)....T(cid:20)..(cid:0)"@rb..T(cid:20)...."(cid:0)@b....................... γ(8)21:.......................P(cid:8)(cid:3)aA..(cid:3)..(cid:3)A(cid:3)..P(cid:8)(cid:3)aA..(cid:3)..A(cid:8)....P(cid:8)Aa..A(cid:8)....P(cid:8)a....P(cid:8)..(cid:8)a....r(cid:8)A....(cid:8)aA..(cid:8)A....aA(cid:3)..(cid:3)..(cid:8)aA(cid:3)(cid:3)..A(cid:3)..(cid:3).....................
1231454672378586 Z2 1231456472378568 Z4 1234513674862578
HOMOLOGIES OF MODULI SPACE M2,1 7
γ(8)22:.......................P(cid:8)aA(cid:3)..(cid:3)..A(cid:3)(cid:3)..P(cid:8)Aa(cid:3)..(cid:3)..(cid:8)A....P(cid:8)Aa..A(cid:8)....rP(cid:8)Pa....P(cid:8)..(cid:8)Pa....(cid:8)....(cid:8)Pa..(cid:8)....P(cid:3)a..(cid:3)..(cid:8)P(cid:3)a(cid:3)..(cid:3)..(cid:3)..................... γ(8)23:.......................P(cid:3)(cid:8)aA..(cid:3)..(cid:3)A(cid:3)..P(cid:3)(cid:8)aA..(cid:3)..A(cid:8)....P(cid:8)Aa..A(cid:8)....P(cid:8)aPr....P(cid:8)..(cid:8)aP....(cid:8)A....(cid:8)aPA..(cid:8)A....aPA....(cid:8)aPA..A....................... γ(8)24:.................(cid:2)P(cid:24)J..(cid:2)..(cid:2)J..(cid:2)P(cid:24)..(cid:2)J..(cid:24)....P(cid:24)J..J(cid:24)....P(cid:24)Pr....P(cid:24)..(cid:24)P....(cid:24)J....(cid:24)PJ..(cid:24)..J..P(cid:2)..(cid:2)..(cid:24)PJ(cid:2)..J(cid:2)..(cid:2)...............
1231453674862578 1231452674863578 1234152467586378 Z2
γ(8)25:.......................QrA(cid:17)(cid:1)X..A..(cid:1)..AQ(cid:17)(cid:1)X....A(cid:1)....Q(cid:17)X......AQ(cid:1)(cid:17)X....A(cid:1)..AQ(cid:1)(cid:17)....A(cid:1)....AQ(cid:1)(cid:17)..A(cid:1)......................... γ(8)26:.......................(cid:24)A(cid:17)Xr(cid:1)....A(cid:1)..A(cid:24)(cid:17)X(cid:1)....A(cid:1)....(cid:24)(cid:17)X......(cid:24)(cid:1)(cid:17)AX....(cid:1)A..(cid:1)(cid:17)A....(cid:1)A....(cid:1)A(cid:17)..(cid:1)A......................... γ(8)27:.......................(cid:17)(cid:1)QX(cid:24)Ar....(cid:1)A..(cid:1)(cid:17)QX(cid:24)A....(cid:1)A....(cid:17)QX(cid:24)......(cid:17)QX(cid:24)(cid:1)....(cid:1)..(cid:17)Q(cid:1)....(cid:1)....(cid:17)Q(cid:1)..(cid:1).........................
1234561374857286 1231456374857268 1231456274857368
γ(8)28:...................(cid:8)(cid:10)XrJH(cid:24)....(cid:10)J..(cid:8)XH(cid:24)..(cid:10)(cid:10)JJ......(cid:8)XH(cid:24)......(cid:8)XH(cid:24)T(cid:20)....T(cid:20)..(cid:8)H..T(cid:20)....T(cid:20)..(cid:8)H..T(cid:20)..................... γ(8)29:.......................H!C(cid:1)r..C..C(cid:1)C..H!C(cid:1)..C..H(cid:1)....H(cid:1)!..(cid:1)H....H!(cid:16)....H..H!(cid:16)....H(cid:1)....H!(cid:16)(cid:1)..H(cid:1)....C!(cid:16)(cid:1)..C..H!(cid:16)(cid:1)CC..(cid:1)C..C.....................
1234156247586738 1234562748571368
6. Appendix 3. Schemas of seven-edge glueings.
γ(7)1: ............(cid:2)H(cid:8)B..(cid:2)B..(cid:2)(cid:2)BB..H(cid:8)....................(cid:2)rB..(cid:2)B..(cid:2)(cid:2)BB............. γ(7)2: ............(cid:2)H(cid:8)B..(cid:2)B..(cid:2)(cid:2)BB..H(cid:8)..............@r(cid:0)......(cid:0)@................. γ(7)3: ............(cid:0)@......(cid:0)@....................(cid:0)@r........(cid:0)@...............
12123456475673 12312345647567 12312345675674 Z2
γ(7)4: ............H(cid:8)......H(cid:8)....................(cid:0)@r........(cid:0)@............... γ(7)5: ............(cid:2)H(cid:8)B..(cid:2)B..(cid:2)(cid:2)BB..H(cid:8)..............H(cid:8)r......(cid:8)H................. γ(7)6: ............H(cid:8)......H(cid:8)....................(cid:8)Hr........(cid:8)H...............
12312345657674 12132345647567 12132345657674 Z2
γ(7)7: ................(cid:3)H(cid:8)C..(cid:3)C..(cid:3)C(cid:3)C..H(cid:8)..............(cid:8)......r(cid:8).................... γ(7)8: ................(cid:3)H(cid:8)C..(cid:3)C..(cid:3)C(cid:3)C..H(cid:8)..............(cid:3)rC..(cid:3)C..(cid:3)C(cid:3)C................ γ(7)9: ............(cid:2)(cid:8)B(cid:24)..(cid:2)B..(cid:2)B(cid:2)B..(cid:8)(cid:24)........(cid:8)(cid:24)......(cid:24)(cid:8)(cid:2)Br..(cid:2)B..(cid:2)B(cid:2)B.............
12313456427567 12312456437567 12341245637567 Z2
8 YURYKOCHETKOV
γ(7)10:................(cid:8)H......(cid:8)H..............(cid:8)......(cid:8)(cid:3)rC..(cid:3)C..(cid:3)C..(cid:3)C.............. γ(7)11:............H(cid:2)B..(cid:2)B..(cid:2)(cid:2)BB..H..............(cid:8)......(cid:8)(cid:2)rB..(cid:2)B..(cid:2)(cid:2)BB............. γ(7)12:................(cid:8)C(cid:3)H..C(cid:3)..C(cid:3)C(cid:3)..(cid:8)H..............rH......H....................
12341245653767 12341245675367 12132456437567
γ(7)13:................H(cid:8)......H(cid:8)..............(cid:8)Hr......(cid:8)H.................... γ(7)14:................C(cid:3)H..C(cid:3)..C(cid:3)C(cid:3)..H..............rH(cid:8)......(cid:8)H.................... γ(7)15:................C(cid:3)H..C(cid:3)..C(cid:3)C(cid:3)..H..............Hr......H(cid:3)C..(cid:3)C..(cid:3)C..(cid:3)C..............
12134245653767 Z2 12134245675367 12314235675467 Z2
γ(7)16:................"(cid:10)Ja....(cid:10)J.."a..(cid:10)J(cid:10)J......"a......(cid:8)"a......(cid:8)r"a.................... γ(7)17:............!(cid:0)@a......!(cid:0)@a........!a......(cid:10)Jr!a....(cid:10)J..!a..(cid:10)(cid:10)JJ............... γ(7)18:................"(cid:0)a......"(cid:0)a........(cid:16)"a......r(cid:1)(cid:16)A"a....(cid:1)A..(cid:1)A(cid:16)"a....(cid:1)A................
12341456275367 12341256375467 12345125637467
γ(7)19:................(cid:8)(cid:10)JP....(cid:10)J..(cid:8)P..(cid:10)(cid:10)JJ......(cid:8)P......(cid:8)P(cid:16)......Ar(cid:8)P(cid:16)(cid:1)....A(cid:1)....(cid:8)AP(cid:1)(cid:16)..A(cid:1)................ γ(7)20:................(cid:8)@P......(cid:8)@P........(cid:8)P......(cid:16)(cid:8)P......rA(cid:16)(cid:1)(cid:8)P....A(cid:1)....A(cid:1)(cid:16)(cid:8)P..A(cid:1)................ γ(7)21:................!(cid:10)Jb....(cid:10)J..!b..(cid:10)(cid:10)JJ......!b......Hr!b......H!b....................
12345125637647 12345125673647 12134256375467
γ(7)22:..................."(cid:0)b......"(cid:0)b........"b......(cid:8)Hr"b......(cid:8)H"b....................... γ(7)23:..................."b(cid:1)A....(cid:1)A.."(cid:1)bA....(cid:1)A...."b......"Hb(cid:8)r......"(cid:8)Hb....................... γ(7)24:..................."@b......"@b........"b......(cid:8)Hr"b......(cid:8)H"b.......................
12134525637467 12134525637647 12134525673647
γ(7)25:................(cid:16)(cid:0)H......(cid:16)(cid:0)H........(cid:16)H......rP(cid:16)H......AP(cid:1)(cid:16)H....A(cid:1)....A(cid:1)P(cid:16)H..A(cid:1)................ γ(7)26:................(cid:16)(cid:10)HJ....(cid:10)J..(cid:16)H..(cid:10)(cid:10)JJ......(cid:16)H......(cid:16)HPr......A(cid:16)HP(cid:1)....A(cid:1)....(cid:16)HA(cid:1)P..A(cid:1)................ γ(7)27:................(cid:16)@H......(cid:16)@H........(cid:16)H......(cid:16)rPH......A(cid:16)P(cid:1)H....A(cid:1)....A(cid:1)(cid:16)PH..A(cid:1)................
12314523647567 12314523647657 12314523674657
HOMOLOGIES OF MODULI SPACE M2,1 9
γ(7)28:..................."Ab....A.."b........"b(cid:16)Pr......L"b(cid:16)P(cid:12)..L(cid:12)....L(cid:12)"b(cid:16)P..L(cid:12)..LL(cid:12)(cid:12)................... γ(7)29:..................."b......"b........(cid:16)Pr"b......L(cid:16)P(cid:12)"b..L(cid:12)....L(cid:12)(cid:16)P"b..L(cid:12)..LL(cid:12)(cid:12)................... γ(7)30:..................."bA....A.."b........"bPr(cid:16)......L"b(cid:16)P(cid:12)..L(cid:12)....L(cid:12)"b(cid:16)P..L(cid:12)..LL(cid:12)(cid:12)...................
12314562364757 12314562367457 12314562367475
γ(7)31:...................b#A(cid:1)....A(cid:1)..bA#(cid:1)....A(cid:1)....b#......b(cid:8)#......rb(cid:8)#........................ γ(7)32:................(cid:8)(cid:10)JH....(cid:10)J..(cid:8)H..(cid:10)(cid:10)JJ......(cid:8)H......(cid:8)HrA(cid:1)....A(cid:1)..(cid:8)HA(cid:1)....A(cid:1)....(cid:8)H.................. γ(7)33:...................(cid:17)H(cid:10)....(cid:10)..(cid:17)H..(cid:10)(cid:10)......(cid:17)(cid:24)H......Lr(cid:17)(cid:24)H(cid:12)..L(cid:12)....L(cid:12)(cid:17)(cid:24)H..L(cid:12)..LL(cid:12)(cid:12)....(cid:17)(cid:24)H.....................
12345156273647 12345126374657 12345612637457
γ(7)34:...................(cid:17)B(H..B..BB..(cid:17)(H........(cid:17)(H(cid:24)......(cid:17)r(H(cid:24)L(cid:12)..L(cid:12)....L(cid:12)(cid:17)(H(cid:24)..L(cid:12)..LL(cid:12)(cid:12)....(cid:17)(H(cid:24)..................... γ(7)35:...................cA(cid:1)"....A(cid:1)..Ac(cid:1)"....A(cid:1)....c"......rcH"......cH"....................... γ(7)36:......................#c(cid:1)....(cid:1)..#c(cid:1)....(cid:1)....#c......#rc(cid:8)H......#c(cid:8)H...........................
12345612673475 12134526374657 12134562637457
γ(7)37:......................cA#....A..cA#....A....c#......rHc(cid:8)#......(cid:8)Hc#........................... γ(7)38:...................@@`"......(cid:20)@@`"....(cid:20)....@D`"(cid:20)D..D..D(cid:20)DD..@r@`"..(cid:20)....@@`"....................... γ(7)39:................@(cid:0)@(cid:0)......@(cid:0)........@(cid:0)......@(cid:0)(cid:0)@r....................
12134562673475 12314567237456 Z2 12134356267475Z2
γ(7)40:................@(cid:0)H(cid:0)......(cid:0)H(cid:0)A....A....H(cid:0)A(cid:0)..A....H(cid:0)(cid:0)@r.................... γ(7)41:...................(cid:17)H(cid:1)(cid:8)H....(cid:1)..(cid:17)(cid:1)H(cid:8)DH..D..D(cid:1)D..DD..(cid:17)H(cid:8)H......(cid:17)H(cid:8)H......r(cid:17)H........(cid:17)H..................... γ(7)42:................(cid:10)@(cid:8)@....(cid:10)..@(cid:8)@..(cid:10)(cid:10)......(cid:8)@......(cid:8)r@....................
12134536267457Z2 12342563751467 12341563752467
γ(7)43:...................(cid:10)J....(cid:10)J....(cid:10)J(cid:10)J............(cid:0)......(cid:0)r....................... γ(7)44:................(cid:0)@......(cid:0)@........(cid:0)r@(cid:0)@......(cid:0)@......(cid:0)@.................... γ(7)45:................H(cid:17)`......H(cid:17)`........H(cid:17)`(cid:0)r@......`(cid:0)@......`(cid:0)@....................
12341356475267 12345136472567 Z2 12345136476256
10 YURYKOCHETKOV
γ(7)46:...................(cid:10)J....(cid:10)J....(cid:10)J(cid:10)J............(cid:10)Jr....(cid:10)J....(cid:10)(cid:10)JJ..................... γ(7)47:................(cid:2)@(cid:8)@..(cid:2)..(cid:2)(cid:2)..@(cid:8)@(cid:8)........(cid:8)r@(cid:8)@......@(cid:8)@(cid:2)..(cid:2)..(cid:2)(cid:2)................ γ(7)48:................@(cid:8)@......@(cid:8)@........(cid:8)@r@......@@(cid:2)..(cid:2)..(cid:2)(cid:2)................
12314356475267 12314536476257 Z2 12314536746257
γ(7)49:................!S....S..!....S....!r(cid:17)PS......!(cid:17)P....A..!(cid:17)PA....A................ γ(7)50:................@@......@@........@@r......@@.................... γ(7)51:...................Q(cid:0)......Q(cid:0)........Qr...................................
12314256475367 12314526746357 Z2 12345263746157
γ(7)52:...................(cid:10)rJ....(cid:10)J..(cid:28)..(cid:10)(cid:10)JJ....(cid:28)......(cid:28)..@..(cid:28)....@..(cid:28).....................
12345136475627
7. Appendix 4. Schemas of six-edge glueings.
γ(6)1: ................H(cid:1)(cid:8)A....(cid:1)A..(cid:1)H(cid:8)A....(cid:1)A....H(cid:8)............r(cid:1)A....A(cid:1)....A(cid:1)..A(cid:1)................ γ(6)2: ................(cid:10)J....(cid:10)J....(cid:10)(cid:10)JJ..................A(cid:1)r....A(cid:1)....A(cid:1)..A(cid:1)................ γ(6)3: ................Z(cid:26)......Z(cid:26)....................rA(cid:1)....A(cid:1)....A(cid:1)..A(cid:1)................
121234536456 121234564563 121234546563
γ(6)4: ................(cid:10)J....(cid:10)J....(cid:10)J(cid:10)J............(cid:10)Jr....(cid:10)J....(cid:10)J(cid:10)J.................. γ(6)5: ................Z(cid:26)......Z(cid:26)..............(cid:10)rJ....(cid:10)J....(cid:10)(cid:10)JJ.................. γ(6)6: ................Z(cid:26)......Z(cid:26)..............(cid:26)Zr......(cid:26)Z....................
123123456456 Z2 123123454656 121323454656 Z2
γ(6)7: ................(cid:0)H(cid:0)......(cid:0)H(cid:0)A....A....H(cid:0)A(cid:0)..A....H(cid:0)@r(cid:0).................... γ(6)8: ................(cid:1)A....(cid:1)A..(cid:1)A....(cid:1)A..........(cid:17)(cid:19)......r.................... γ(6)9: ............................(cid:16)(cid:17)........(cid:19)......r....................
121345326456 123134526456 123134542656