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Homotopy types of moment-angle complexes
based on joint work with Jelena Grbic, Stephen Theriault and Jie Wu

Taras Panov
Lomonosov Moscow State University

The 39th Symposium on Transformation Groups Tokyo University of Science, Japan, 23­25 November 2012

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Homotopy typ es of m-a complexes

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1. Problems

identifying the homotopy type of the moment-angle complex ZK for certain simplicial complexes K; describing the multiplication and higher Massey products in the Tor-algebra H (ZK ) = Tork[v1 ,...,vm ] (k[K], k) of the face ring k[K]; describing the Yoneda algebra Extk[K] (k, k) in terms of generators and relations; describing the structure of the Pontryagin algebra H (DJ (K)) and its commutator subalgebra H (ZK ) via iterated and higher Whitehead (Samelson) products; identifying the homotopy type of the loop spaces DJ (K) and ZK .

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Homotopy typ es of m-a complexes

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2. Preliminaries
(X , A) a pair of spaces. K a simplicial complex on [m] = {1, 2, . . . , m}, Given I = {i1 , . . . , ik } [m], set (X , A)I = Y1 â · · · â Ym where Yi = X A if i I , if i I . / K.

The K-polyhedral product of (X , A) is (X , A)K =
I K

(X , A)I =
I K i I

Xâ
i I /

A.

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Example
1

(X , A) = (D 2 , S 1 ), ZK := (D 2 , S 1 )K the moment-angle complex. It has an action of the torus T m . (X , A) = (CP , pt ), DJ (K) := (CP , pt )K the Davis­Januszkiewicz space. (X , A) = (C, Câ ), U (K) := (C, Câ )K = Cm
{i1 ,...,ik }K /

2

3

{zi1 = · · · = zik = 0}

the complement of a coordinate subspace arrangement.

Theorem
There exists a deformation retraction Z
Taras Panov (MSU)

K

U (K) - Z

K
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Homotopy typ es of m-a complexes


There exists a homotopy fibration Z
K

-

D J (K ) (CP , pt )
K

-

(CP )

m

(D 2 , S 1 )K which splits after looping:

(CP , CP )K

DJ (K)

ZK â T

m

Warning: this is not an H -space splitting

Proposition
There exists an exact sequence of noncommutative algebras 1 - H ( ZK ) - H ( DJ (K)) - [u1 , . . . , um ] - 1 where [u1 , . . . , um ] denotes the exterior algebra and deg ui = 1.
Taras Panov (MSU) Homotopy typ es of m-a complexes Tokyo 24 Nov 2012 5 / 15

Ab


Let k denote Z or a field. The face ring (the Stanley­Reisner ring) of K is given by k[K] := k[v1 , . . . , vm ] where deg vi = 2. vi1 · · · vik = 0 if {i1 , . . . , ik } K /

Theorem
H (DJ (K)) k[K] = Extk[K] (k, k) H ( DJ (K)) = H (ZK ) Tork[v1 ,...,vm ] (k[K], k) = H [u1 , . . . , um ] k[K], d , = =
I K

k is a field dui = vi , dvi = 0 KI = K|
I

H

-|I |-1

(KI )

k[K] is a Golod ring if the multiplication and all higher Massey operations in Tork[v1 ,...,vm ] (k[K], k) are trivial.
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3. The case of a flag complex
A missing face of K is a subset I [m] such that I K, but every proper / subset of I is a simplex of K. K is a flag complex if each of its missing faces has two vertices. Equivalently, K is flag if any set of vertices of K which are pairwise connected by edges spans a simplex. {flag complexes on [m]} {simple graphs on [m]} K K() K
1 1-1

(one-skeleton)

where K() is the clique complex of (fill in each clique of with a simplex).
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A graph is chordal if each of its cycles with

4 edges has a chord. 4.

Equivalently, is chordal if there are no induced cycles of length

Theorem (Fulkerson­Gross)
A graph is chordal if and only if its vertices can be ordered in such a way that, for each vertex i , the lesser neighbours of i form a clique. (perfect elimination ordering)

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Theorem (Grbic-P-Theriault-Wu)
K is flag, k a field. The following are equivalent:
1 2 3 4

k[K] is a Golod ring; the multiplication in H (ZK ) is trivial; = K1 is a chordal graph; ZK has homotopy type of a wedge of spheres.

The equivalence (1) (2) (3) was proved by Berglund and J¨ ollenbeck in 2007. Implications (1) (2), (2) (3) and (4) (1) are valid for arbitrary K. However, (3) (4) fails in the non-flag case.

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Example
5 B ¨ ¨ rr ¨ rr ¨ j r6 4¨ T e ¡e ¡ 2e e ¡ ¡ e ¡e ¡ e¡ e¡ 1e¡ 6 r e¡3 4 ¨c rrd d ¨ rr¨ ¨ ¨ d% 5

The nonzero cohomology groups of ZK are H 0 = Z, H 5 = Z10 , H 6 = Z15 , H 7 = Z6

H 9 = Z/2. All Massey products vanish for dimensional reasons, so K is Golod (over any field).

Nevertheless, ZK is not homotopy equivalent to a wedge of spheres because of the torsion. In fact, Z
K

(S 5 )

10

(S 6 )

15

(S 7 )

6

7 RP 2 .

Question
Assume that H (ZK ) has trivial multiplication, so that K is Golod, over any field. Is it true that ZK is a co-H -space, or even a suspension, as in the previous example?
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How many spheres in the wedge? What if ZK is not a wedge of spheres?

Theorem
For any flag complex K, there is an isomorphism H DJ (K) T u1 , . . . , um = ui2 = 0, ui uj + uj ui = 0 for {i , j } K

where T u1 , . . . , um is the free algebra on m generators of degree 1. Remember the exact sequence of non-commutative algebras 1 - H ( ZK ) - H ( DJ (K)) - [u1 , . . . , um ] - 1
Ab

Proposition
For any flag complex K, the Poincar´ series of H (ZK ) is given by e P H (ZK ); t = (1 + t )
m -n

1 , (1 - h1 t + · · · + (-1)n hn t n )

where h(K) = (h0 , h1 . . . , hn ) is the h-vector of K.
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Theorem
Assume that K is flag. The algebra H (ZK ), viewed as the commutator subalgebra of H (DJ (K)), is multiplicatively generated by 0 I [m] dim H (KI ) iterated commutators of the form [uj , ui ], [uk1 , [uj , ui ]], ..., [uk1 , [uk2 , · · · [ukm-2 , [uj , ui ]] · · · ]]

where k1 < k2 < · · · < kp < j > i , ks = i for any s , and i is the smallest vertex in a connected component not containing j of the subcomplex K{k1 ,...,kp ,j ,i } . Furthermore, this multiplicative generating set is minimal.

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Here is an important particular case (corresponding to K = m points). It is an analogue of the description of a basis in the commutator subalgebra of a free algebra, given by Cohen and Neisendorfer:

Lemma
Let A be the commutator subalgebra of T u1 , . . . , um /(ui2 = 0): 1 - A - T u1 , . . . , um /(ui2 = 0) - [u1 , . . . , um ] - 1 where deg ui = 1. Then A is a free associative algebra. It is minimally generated by the iterated commutators of the form [uj , ui ], [uk1 , [uj , ui ]], ..., [uk1 , [uk2 , · · · [ukm-2 , [uj , ui ]] · · · ]]
m

where k1 < k2 < · · · < kp < j > i and ks = i for any s . The number of commutators of length
Taras Panov (MSU)

is equal to ( - 1)

.
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Homotopy typ es of m-a complexes

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Corollary
Assume that K is flag and ZK has homotopy type of a wedge of spheres. Then the number of spheres of dimension + 1 in the wedge is given by 0 m. |I |= dim H (KI ), for 2 In particular, H i (KI ) = 0 for i > 0 and all I .

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Homotopy typ es of m-a complexes

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Reference

Jelena Grbic, Taras Panov, Stephen Theriault and Jie Wu. Homotopy types of moment-angle complexes. Preprint (2012); arXiv:1211.0873.

Taras Panov (MSU)

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