On the cohomology of polyhedral products with space pairs $(D^1, S^0)$

0.15 MB·English

by  Li Cai

ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE
PAIRS (D1,S0)
3
LI CAI
1
0
2
Abstract. In this note we prove that H(RZK), namely the integer cohomology ring of
n polyhedral products with space pairs (D1,S0), can also be described explicitly by a multi-
a
J plicative Hochster’s formula.

0
1
Theobjective ofthisnoteistounderstandH(RZ ), inawayparallellytotheworkrelated
] K
T
tothecohomologyofmoment-angle complexesZ (seee.g. [3], [2], [4])byBuchstaber, Panov
K
A
and Baskakov.
.
h After some constructions, an explicit formula for H(RZ ) is obtained in Theorem 1.6,
t K
a
the whole proof of which is completed later after Proposition 2.1. Some applications are
m
discussed in Remark 1.8.
[

2
v 1. On the Additive Structure
8
1 In this note let Z be the ring of integers and let all coefficients be from Z.
5
1 For a non-negative integer m, we write [m] for the set {1,··· ,m}. A simplicial complex
.
1 K on [m] means the vertice set of K is identified with [m], such that
0
3 (1) there is an unique emptyset which belongs to every simplex of K;
1
(2) as a subset of [m], σ is a simplex of K implies that any face (subset) of σ is a simplex.
:
v
i For instance, if K contains [m] as a simplex, then every subset σ = {i ,...,i } is a p-simplex
X 0 p
of K.
r
a
The geometric realization of K will be denoted by |K|. By RZ we denote a specific
K
polyhedral product (we follow [3] on this notation) defined as the union
(1) RZ = {(x ,··· ,x ) ∈ (D1)m|x ∈ S0, when i 6∈ σ},
K [ 1 m i
σ∈K
thus RZ can be embedded in the product (D1)m in an obvious way.
K
Example 1.1. It is known that if K is a simplicial (n − 1)-sphere (a simplicial complex
homeomorphic to the (n − 1)-sphere), then RZ is an n-dimensional manifold (see [3, p.
K
98]). Let K be the pentagon on [5], namely it is a 1-dimensional simplicial complex with
1
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 2
edges {i,i+ 1} for i = 1,2,3,4 and {1,5}. For the five 1-simplices (resp. five 0-simplices)
of K, let each 1-simplex (resp. 0-simplex) correspond to the 23 squares (resp. the 24 edges)
(D1)2 ×(S0)3 (resp. D1 ×(S0)4) and let (S0)5 be the 25 vertices as in definition (1). Then
by counting the Euler number we can identify this RZ as a surface with genus 5.
K
1.1. A Cell Decomposition. Now we give (D1)m a CW structure by a simplicial decom-
position to each component D1 (i = 1,...,m), which descends to RZ as a subcomplex: let
i K
01 bethe1-simplex and0 , 1 betwo 0-simplices of theunit interval [0,1] ∼= D1 respectively,
i i i i i
then each oriented cell of (D1)m can be written as
e ×···×e , e = 0 , 1 or 01 .
1 m i i i i
With the decomposition above, we find
Lemma 1.2. A cell e = e ×···×e of (D1)m belongs to its subspace RZ if and only if
1 m K
σ := {i|e = 01 }
e i i
is a simplex of K.
In what follows we will abuse the notation e : it may mean a simplex or the corresponding
i
simplicial chain, and we will declare what it means depending on different situations. For a
topologicalspaceX, wedenoteS(X)(resp. S∗(X))tobeitssingularchains(resp. cochains);
C (X)(resp. C∗(X))willmeanthecellularchains(resp. cochains)whenX isaCW complex,
e e
and C(X) (resp. C∗(X)) will be denoted for the simplicial chains (resp. cochains) when X
is a simplicial complex. The notation H(X) (resp. H∗(X)) means the singular homology
(resp. cohomology) of X.
m
1.2. A Brief Review. LetX = |K |beaproductspacebytopologizedfinitesimplicial
i=1 i
Q
complexes K , note that now each C∗(K ) is finitely generated. Let (⊗m C∗(K );d) be the
i i i=1 i
differential graded abelian group with generators
e∗ ⊗···⊗e∗
1 m
where e∗ is the dual of the simplicial chain generated by e in K , and whose differential d
i i i
satisfies
m
(2) d(e∗ ⊗···⊗e∗ ) = (−1)Pij−=11deg(e∗j)e∗ ⊗···⊗e∗ ⊗de∗ ⊗e∗ ⊗···⊗e∗
1 m X 1 i−1 i i+1 m
i=1
on each generator (onthe right hand side the differential dis the usual simplicial coboundary
operator). For instance, assume that i = 1,2,3, from d0∗ = −01∗, d1∗ = 01∗ and d01∗ = 0,
i i i i i
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 3
one has
d(0∗ ⊗01∗ ⊗1∗) = −01 ⊗01 ⊗1∗ −0∗ ⊗01∗ ⊗01∗.
1 2 3 1 2 3 1 2 3
C∗(X) means the set of cellular cochains generated by dual cells
e
(e ×···×e )∗,
1 m
on which we define a morphism
f : C∗(K )⊗···⊗C∗(K ) −−−→ C∗(X)
1 m e
(3)
e∗ ⊗···⊗e∗ −−−→ (e ×···×e )∗,
1 m 1 m
which is a 1-1 correspondence by a comparison of basis, thus we can equip C∗(X) with a
e
differential d such that f is a cochain map, and it will be denoted as (C∗(X);d).
e
Proposition1.3. ConsiderX = m |K | asa CW complexwith cellularcochains(C∗(X);d),
i=1 i e
Q
and assume that A is a subcomplex of X subject to this cell decomposition, then there is an
isomprphism
H∗(A) ∼= H(C∗(A);d)
e
preserving degrees. For instance, we have isomorphisms
H( Cp1(K )⊗···⊗Cpm(K );d) −−−f→ Hp(C∗(X);d) ∼= Hp(X)
M 1 m ∼= e
Pmi=1pi=p
where p are non-negative integers.
i
A standard cellular (co)homology argument (see e.g. [6, Theorem 57.1, p. 339]) will pro-
vide a proof without involving cup products. We will give full version of this proposition
later (i.e. Proposition 2.1), but here the simplicial cochain algebra of C∗(D1) with simpli-
i
cial cup products will be presented, because it motivates the algebraic construction later.
Henceforth let us work with the following basis
(4) {1 = 1∗ +0∗,t = 1∗,u = 01 }
i i i i i i i
of C∗(D1) rather than the original {0∗,1∗,01∗}. By definition (see e.g. [6, p. 292]), we have
i i i i
the following relations on simplicial cup products:
1 ∪t = t ∪1 , 1 ∪u = u ∪1 , t ∪t = t , u ∪u = 0, t ∪u = 0, u ∪t = u .
i i i i i i i i i i i i i i i i i i
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 4
1.3. An Algebraic Description. In what follows, x will mean the monomial x ...x
σ i0 ip
whenσ = {i ,...,i }isasubset of[m]. ByZ[u ,...,u ;t ,...,t ]wedenotethedifferential
0 p 1 m 1 m
graded free Z-algebra with 2m generators, such that
(5) degu = 1, degt = 0; du = 0, dt = u ,
i i i i i
where the differential d follows the rule (2) on each monomial. Let R be the quotient algebra
of Z[u ,...,u ;t ,...,t ] subject to the following relations
1 m 1 m
(6) u t = u , t u = 0, u t = t u , t t = t , u u = 0, u u = −u u , t t = t t ,
i i i i i i j j i i i i i i i j j i i j j i
for i,j = 1,...,m such that i 6= j, in which the Stanley-Reisner ideal I is defined to be
K
generated by all square-free monomials u such that σ is not a simplex of K.
σ
One can check that the relations (6) listed above are compatible with the differential d,
for instance,
d(t t ) = u t +t u = u t = u = dt .
i i i i i t i i i i
Theorem 1.4. There is a (degree-preserving) ring isomorphism
(7) H(R/I ;d) ∼= H(RZ ).
K K
Proof. Here we only prove the additive part of this theorem, and the whole proof involving
products will be completed later after Proposition 2.1.
We start from a cochain map, which is generated by mapping each square-free monomial
to a tensor product of simplicial cochains, namely
θ : R −−−→ C∗(D1)⊗···⊗C∗(D1 )
i m
x ...x −−−→ e∗ ⊗···⊗e∗ ,
1 m 1 m
in which (we are using the basis (4))
1 , if x = 1,
 i i
e∗ = t , if x = t ,
i  i i i
u , if x = u .
 i i i


The additive generators of R are square-free monomials follows from the relations (6), and
by definition θ induces an isomorphism between abelian groups such that d◦θ = θ◦d. Then
from the following diagram about cochain maps (in which q is the quotient and i∗ is induced
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 5
by the inclusion)
R −−f−◦θ→ C∗(D1 ×···×D1 )
∼= e 1 m
(8) q i∗
 
R/yI −−−→ C∗(RyZ ),
K ∼= e K
together with Lemma 1.2, we observe that the quotient part in the Stanley-Reisner ideal
corresponds to the cells not contained in RZ , thus by Proposition 1.3, this part of proof is
K
(cid:3)
completed.
1.4. A Combinatorial Description (Hochster’s Formula). With the notation x , we
σ
can write each square-free monomial of R/I in the form
K
u t , σ ∩τ = ∅, σ ∈ K
σ τ
(for instance, u t u t t = u t ).
2 3 4 5 6 2,4 3,5,6
When ω is a subset of [m] (may be empty), we denote K to be the full subcomplex of K
ω
in ω, namely it is the subcomplex
{σ ∩ω|σ ∈ K},
and R/I | will be denoted as a subgroup of R (as a group) generated by
K ω
{u t |σ ∈ K s.t. σ ⊔τ = ω, σ ∩τ = ∅}.
σ τ
We observe that R/I | is closed under the differential d and R/I = R/I | ,
K ω K ω⊂[m] K ω
L
therefore
∼
H(R/I ;d) = H(R/I | ;d).
K M K ω
ω⊂[m]
Each ω ⊂ [m] yields a 1-1 cochain map (on the right hand side we use the usual simplicial
coboundary operator)
λ : R/I | −−∼=−→ C∗(K )
ω K ω ω
(9)
u t −−−→ σ∗,
σ τ
which induces an additive isomorphism of cohomology
Hp(R/I | ) −−λ−ω→ Hp−1(K ),
K ω ω
∼=
e
because
m
d(t ...t ) = t ...t u t ...t
1 m X 1 i−1 i i+1 m
i=1
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 6
behaves as the augmentation and λ∗ maps H0(R) isomorphically onto H−1(∅).
∅
e
Remark 1.5. Indeed, λ induces cochain maps since every time we jump over an index in
σ, we will get a −1. Here the construction of λ is motivated by Baskakov’s idea in [1].
Therefore we can equip Hp−1(K ) with a product structure described by simpli-
ω⊂[m] ω
L
cial cochains, such that cochain maep λ induces an isomorphism between cochain algebras.
Together with Theorem 1.4, we have the following theorem, the full proof of which will be
completed later.
Theorem 1.6. There are isomorphisms
Hp−1(K ) −−∼=−→ Hp(R/I ) −−∼=−→ Hp(RZ )
M ω λ−1 K K
ω⊂[m] e
for all positive integers p. Moreover, after endowing Hp−1(K ) the product structure
ω⊂[m] ω
L
induced by λ, they are isomorphisms between Z-algebras. e
Thus the cohomology ring of RZ will be clear once we write down all cochains repre-
K
senting the (reduced) simplicial cohomology for each K .
ω
Example 1.7. Assume that K is the pentagon described in Example 1.1, let us compute the
cohomology of RZ . First we can write down all the full subcomplexes of K with non-trivial
K
cohomology, namely
1 :H−1(K )|1;
∅
e
α −α :H0(K )|u t ,H0(K )|u t ,H0(K )|u t ,H0(K )|u t ,H0(K )|u t ;
1 5 1,3 1 3 1,4 1 4 2,4 2 4 2,5 2 5 3,5 3 5
e e e e e
β −β :H0(K )|u t ,H0(K )|u t ,H0(K )|u t ,H0(K )|u t ,H0(K )|u t ;
1 5 1,3,4 1 3,4 2,4,5 2 4,5 3,5,1 3 1,5 4,1,2 4 1,2 5,2,3 5 2,3
e e e e e
γ :H1(K )|u t ,
1,2,3,4,5 1,2 3,4,5
e
in which we write a generator for each group, via λ. Thus (by Theorem 1.6) H(RZ ) has
K
ten generators in degree 1 (i.e. α , β , i = 1,...,5), and one generator in degree 0 (i.e. 1 the
i i
unit) and degree 2 (i.e. γ) respectively. Note that these generators may not be unique: for
instance,
d(u t ) = −u t −u t +terms equivalent to 0,
1 2,3,4,5 1,2 3,4,5 1,5 2,3,4
and in the same way −u t is equivalent to u t +u t . A little computation shows, up
1 3,4 3 1,4 4 1,3
to equivalences in cohomology, all non-trivial cup products are listed as follows (where we
omit the notation ∪)
γ = α β = α β = α β = α β = α β = β β = β β = β β = β β = β β .
1 2 2 5 3 3 4 1 5 4 1 2 2 3 3 4 4 5 5 1
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 7
We can use the construction (8) in the proof of Theorem 1.4 to identify all corresponding
cellular cochains.
Remark 1.8. From Theorem 1.6, after a comparison with the result in [3], it follows that
there is an additive isomprphism between H(Z ) and H(RZ ) if we forget about degrees.
K K
For instance, we can also find arbitrary torsions in H(RZ ) when K are special simplicial
K
spheres (see [5, Theorem 11.11]). As algebras, the difference between H(Z ) and H(RZ )
K K
comes from defferent sign rules and square rules.
2. On the Cup Product
In this section we will describe cup products in Proposition 1.3, and then finish the proofs
of Theorem 1.4 and Theorem 1.6.
Assume that X and Y are two topological spaces, we define a map g : S∗(X)⊗S∗(Y) →
Hom(S(X)⊗S(Y),Z) generated by
(10) g(c∗ ⊗c∗)(c′ ⊗c′ ) = c∗ (c′ )c∗ (c′ ), c∗ ∈ S∗(X), c∗ ∈ S∗(Y),
X Y X Y X X Y Y X Y
(the evaluation is trivial when degrees do not match) for each singular trangles c′ and c′ .
X Y
ByEilenberg-Zilber theorem, there is a natural chain equivalence µ between singular chain
complexes (unique up to chain homotopy, see [8]; the defferential on tensors follows the rule
(2))
S(X ×Y) −µ−X−×→Y S(X)⊗S(Y),
∼=
and the cup product is defined via the following
S∗(X)⊗S∗(X) −µ−∗X−×−X−◦→g S∗(X ×X) −−d−∗→ S∗(X)
in which d∗ is induced by the diagonal map X → X × X (the cross product is defined by
µ∗ ◦g).
X×X
From now on let Topm be the category with m-products of toplogical spaces as objects,
whose morphisms are m-products of continuous maps, and let C be the category of chain
complexes equiped with chain maps.
Consider X = m |K | as an object of Topm: a product by finite topologized simplicial
i=1 i
Q
complexes.
Thusinthefollowing diagram, allmorphisms canbeconsidered asnaturaltransformations
between functors from Topm to C, where d is the diagonal chain map from S(|K |) to
i i
S(|K |)⊗S(|K |), and µ is defined by using µ for m−1 times, factoring out one component
i i
e
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 8
each time:
(11) S( m |K |)⊗S( m |K |)µeX⊗µeX// ⊗m S(|K |)⊗⊗m S(|K |)
i=1 i i=1 i i=1 i i=1 i
❥❥❥❥µ❥X❥×❥X❥❥◦❥d❥Q❥❥❥❥❥44 Q OO
m
S(Qi=1|Ki|)❯❯❯❯❯❯❯❯µe❯X❯❯❯❯❯❯❯ T
S(|K |**)⊗···⊗S(|K ⊗|)mi=1µ|Ki|×|Ki|◦// d⊗i m S(|K |)⊗S(|K |),
1 m i=1 i i
inwhichT isdefinedasfollows. Foreachtensorbysingulartriangles⊗ c ⊗c′ ∈ ⊗m S(|K |)⊗
i i i i=1 i
S(|K |),
i
T(c ⊗c′ ⊗···⊗c ⊗c′ ) = (−1)ε(c,c′)c ⊗···⊗c ⊗c′ ⊗···⊗c′
1 1 m m 1 m 1 m
where the mod 2 invariant ε(c,c′) is generated by this rule: each operation of changing
the order of any adjacent two terms will produce a multiplication of their degrees, thus we
obtain ε(c,c′) by summing up these multiplications for an arbitrary sequence of operations,
for instance one choice for ε(c,c′) can be
m
ε(c,c′) = degc′ degc .
X iX j
i=1 j>i
It is straightforward to check that T is a chain map. From the method of acyclic model with
the model category as m-products of topologized simplices (see [7], the case for 2-products
can be found in [9, p. 252]), it follows that there is a natural chain homotopy between
µ ⊗µ ◦µ ◦d and T ◦(⊗m µ ◦d )◦µ .
X X X×X i=1 |Ki|×|Ki| i X
e e e
Thereforeafter takingHomtodiagram(11), we obtainthefollowing (whereg , g aredefined
1 2
from (10))
⊗m C∗(K )⊗⊗m C∗(K ) −T−e∗−:=−g−2−−1◦−T−∗−◦→g1 ⊗m C∗(K )⊗C∗(K )
i=1 i i=1 i i=1 i i
g1 g2
 
Hom(⊗m S(K )y⊗⊗m S(K ),Z) −−T−∗→ Hom(⊗m S(Ky)⊗S(K ),Z),
i=1 i i=1 i i=1 i i
in which T∗ is well defined since Im(T∗ ◦g ) ⊂ Im(g ), and here every C∗(K ) is considered
1 2 i
as a subgeroup of S∗(|K |) (i = 1,...,m), in which only simplicial chains are non-trivially
i
evaluated.
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 9
Proposition 2.1. Assume that e∗ = ⊗m e∗ and e∗′ = ⊗m e∗′ are tensors of dual simplices
i=1 i i=1 i
in (⊗m C∗(K ) ⊂)⊗m S∗(|K |), then as elements in S∗( m |K |) on both sides, we have
i=1 i i=1 i i=1 i
Q
(12) µ∗ ◦g(⊗m e∗)∪µ∗ ◦g(⊗m e∗′) = (−1)ε(e∗,e∗′)µ∗ ◦g(⊗m e∗ ∪e∗′),
X i=1 i X i=1 i X i=1 i i
e e m e
up to a cochain homotopy. Moreover, consider X = |K | as a CW complex whose cells
i=1 i
Q
are m-products of (topologized) simplices with i-th component in K , and suppose that A is
i
a subcomplex of X with e∗ and e∗′ in C∗(A), then as elements in S∗(A), (12) holds up to a
e
cochain homotopy.
Proof. After a little chasing on the diagram (11), the first statement follows from
µ∗ ◦g(⊗m e∗)∪µ∗ ◦g(⊗m e∗′) = d∗ ◦µ∗ ◦µ∗ ⊗µ∗ ◦g (⊗m e∗ ⊗⊗m e∗′)
X i=1 i X i=1 i X×X X X 1 i=1 i i=1 i
e e = µ∗ ◦(⊗m µe e◦d )∗ ◦T∗ ◦g (⊗m e∗ ⊗⊗m e∗′)
X i=1 |Ki|×|Ki| i 1 i=1 i i=1 i
= (e−1)ε(e∗,e∗′)µ∗ ◦(⊗m µ ◦d )∗ ◦g (⊗m e∗ ⊗e∗′)
X i=1 |Ki|×|Ki| i 2 i=1 i i
= (−1)ε(e∗,e∗′)µe∗ ◦g(⊗m e∗ ∪e∗′),
X i=1 i i
e
in which from the second line to the third we use the (co)chain homotopy and change T∗ for
the sign (−1)ε(e∗,e∗′). Now we prove the part about the subcomplex A. Denote all ceells of
A by E , we can consider E as a subcategory of Topm with inclusions as morphisms, then
A A
there is a chain equivalence (see e.g. [6, Theorem 31.5], together with the fact that each e
in E is a neighborhood deformation retract of A = e)
A Se∈EA
m
∼
S(A) = colim S( e )
e∈EA Y i
i=1
m
where e = e with each simplex e in K . Next, from the naturality of µ we have the
i=1 i i i
Q
diagram
S(X) −−µe−X→ ⊗mi=1S(|Ki|)
x x
 
S(A) ∼= colim S( m e ) −µe−X−|→A colim ⊗m S(e ),
e∈EA Qi=1 i e∈EA i=1 i
to which after taking Hom, it is not difficult to observe
C∗(X) ∼= (⊗m C∗(K )) −µe−∗X−◦→g S∗(X)
e i=1 i
 
 
C∗y(A) −µe−∗X−|A−◦−g→A S∗(yA).
e
ON THE COHOMOLOGY OF POLYHEDRAL PRODUCTS WITH SPACE PAIRS (D1,S0) 10
where g : C∗(A) → Hom(colim ⊗m S(e ),Z) is defined by evaluating each e˜∗ ∈ C∗(A)
A e e∈EA i=1 i e
non-trivially on its dual e˜ ∈ colim ⊗m S(e ) only. Then we can replace everything of
e∈EA i=1 i
diagram (11) by colimits, and repeat the process for X. (cid:3)
It is not difficult to find the sign rule for commutativity defined by ε coincides with the
one in R/I , which is defined in (6). Therefore we have completed all proofs.
K
References
[1] I. Baskakov, Cohomology of K-powers of spaces and the combinatorics of simplicial divisions, Russian
Math. Surveys 57 (2002), no. 5, 9899990.
[2] I. Baskakov, V. Buchstaber and T. Panov, Algebras of cellular cochains, and torus actions, Russian
Math. Surveys 59 (2004), no. 3, 5625563.
[3] V.BuchstaberandT.Panov,Torus actions and their applications in topology and combinatorics ,AMS
University Lecture Series, volume 24, (2002).
[4] T. Panov, Cohomology of face rings and torus actions, arXiv:math.AT/0506526.
[5] Fr´ed´ericBosioand LaurentMeersseman, Real quadrics in Cn, complex manifolds and convex polytopes,
Acta Mathematica 197 (2006), no. 1, 53-127. MR 2285318
[6] J. Munkres, Elements of algebraic topology, the Benjamin/Cummings Publishing Company, California,
1984.
[7] S. Eilenberg and S. MacLane, Acyclic models, American Journal of Mathematics, vol. 75 (1953), pp.
189-199.
[8] , On the groups H( ,n), I, Annals of Mathematics, vol. 88, No. 1, (1953), pp. 55-106.
Q
[9] E. Spanier, Algebraic topology, Springer-Verlag,New York, 1966.
Graduate School of Mathematics, Kyushu University, 744, motooka, nishi-ku, Fukuoka-
city 819-0395, Japan
E-mail address: [email protected]