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Mo del categories and homotopy colimits in toric top ology
Taras Panov Moscow State University

joint work with Nigel Ray

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1. Motivations. Object of study of "toric top ology": torus actions on manifolds or complexes with a rich combinatorial structure in the orbit quotient. Particular examples:

ћ Non-singular compact toric varieties M 2n
T n-action is a part of an algebraic C n -action with a dense orbit;

ћ (Quasi)toric manifolds M 2n of Davis-Januszkiewicz
"lo cally standard" (i.e., lo cally lo ok like T
n

acting on Cn )

and M /T combinatorially is a simple p olytop e;

ћ Torus manifolds of Hattori-Masuda, "momentangle complexes" , complex co ordinate subspace arrangement complements etc.

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2. Simplicial complexes and face rings. K a simplicial complex on V = {v1, . . . , vm} (e.g., the dual to the b oundary of a simplicial p olytop e). K a simplex. R[v1, . . . , vm] p olynomial algebra on V over R, deg vi = 2. Given V , set v := i vi. The Stanley-Reisner algebra (or face ring) of K is R[K ] := R[v1, . . . , vm]/(v : K ). / Ex 1.
s ff ??d d ?f d ? f d ? f d ? f d s fs Е? rr d Е ЕЕ rr d ЕЕ rrd Е rd Е rs d r Е s

2

K:

4

5

1

3

R[K ] = R[v1, . . . , v5]/(v1v5, v3v4, v1v2v3, v2v4v5).
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The Poincar? series of R[K ] is given by e F (R[K ]; t) = fit2(i+1) 2 i+1 i=-1 (1 - t )
n -1

h0 + h1t2 + . . . + hnt2n = , (1 - t2)n where dim K = n - 1, fi is the numb er of idimensional simplices in K , f-1 = 1, and the numb ers hi are defined from the second identity. A missing face of K is a subset V s.t. K , / but every prop er subset of is a simplex. K is a flag complex if any of its missing faces has two vertices. In this case R[K ] = T (v1, . . . , vm)
/(

vivj - vj vi = 0 for {i, j } K,

vivj = 0 for {i, j } K , / a quadratic algebra.

)

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3. Sample questions. [g-conjecture] Characterise the f -vectors (f0, . . . , fn-1) of triangulations of S n-1 (done for p olytop es). [Charney-Davis conj] Let K 2q-1 b e flag Gorenstein* (e.g., a sphere triangulation). Then (-1)q (h0 - h1 + h2 - h3 + . . . + h2q ) 0.

Calculate the (co)homology of R[K ]. When the Ext-cohomology Extk[K ](k, k) has a rational Poincar? e series?

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The Davis-Januszkiewicz space DJ (K ) :=

K

B T B T m = (CP )m.

Let M 2n b e a toric variety (or a quasitoric manifold) and K n-1 the underlying simplicial complex of the corresp onding fan. Prop 2. DJ (K ) E T n ЧT n M 2n; H (DJ (K ); Z) = HT n (M ; Z) = Z[K ]. Define
) m. ZK := hofibre DJ (K ) B T (

The space ZK is a finite cell complex acted on by T m, called the moment-angle complex. There is a principal T m-n-bundle ZK M . This space also has many other interesting interpretations, e.g. as a complex co ordinate subspace arrangement complement or as a level surface for a certain moment map.
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4. (Co)homology of face rings and toric spaces. Thm 3 (Buchstab er-P). There is an isomorphism of bigraded algebras
, H (ZK ; Z) = TorZ[v ,...,v ](Z[K ], Z) m 1 [ ] H [u , . . . , u ] Z[K ]; d , = m 1

where dui = vi, dvi = 0. What ab out Extk[K ](k, k)? The fibration DJ (K ) B after lo oping: DJ (K ) an H -space splitting, and Pontrjagin homology rings T m with fibre ZK splits ZK Ч T m. This is not the exact sequence of

0 H(ZK ) H( DJ (K )) [u1, . . . , um] 0 do es not split in general. Prop 4. H( DJ (K ), k) = Extk[K ](k, k) Idea of pro of: Use Adams' cobar construction and formality of DJ (K ).
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Prop 5. Supp ose K is flag. Then H( DJ (K ), k) = Tk(u1, . . . , um)
/(

u2 = 0, i

uiuj + uj ui = 0 for {i, j } K . Idea of pro of: Use Koszul duality for algebras. Cor 6. If K is flag then ( DJ (K )) Z Q = FL(u1, . . . , um)
/(

)

[ui, ui] = 0,
)

[ui, uj ] = 0 for {i, j } K , where FL( ) is a free Lie algebra and deg ui = 1. Cor 7. If K is flag, then the rational homology Poincar? series of DJ (K ) is given by e (1 + t)n F H( DJ (K )); t = . nhn tn 1 - h1t + . . . + (-1)
( )

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5. Categories and colimits. cat(K ): face category of K (simplices and incl); mc: a mo del category (e.g., top, tgp or dga); X mc X K : cat(K ) mc exp onential diagram; its value on is the inclusion X X ; X = pt. Many previous constructions are colimits, e.g., DJ (K ) = colimtop B T K , R[K ] = dual coalgebra of R[K ] = colimdgc C (v )K , where C (v ) is the symmetric coalgebra on v , deg v = 2. Cor 8. Assume K is flag. Then DJ (K ) H( DJ (K ), Q) ( DJ (K )) Z Q = colimtgp T K ; = colimga [u]K ; = colimgl CL(u)K ,

where CL(u) is the commutative Lie algebra, deg u = 1. In general colimit mo dels do not work! (Lo ok at K = 2, in which case DJ (K ) is not coformal.)
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6. Homotopy colimit mo dels. Appropriate notions of homotopy colimits exist in the mo del categories tgp, tmon, dga, dgc and dgl. Thm 9. (P.-Ray-Vogt) The lo op space functor : top tmon commutes with the homotopy colimit, i.e., there is a weak equivalence ho colimtop D ho colimtmon D for every diagram D : c top. For diagrams over cat(K ) we get Thm 10. (P.-Ray-Vogt) There is a homotopy commutative diagram
h ho colimtop B T K - K ho colimtgp T K -

p

K



,

DJ (K )

-K -

h

colimtgp T K

in which pK and hK are weak equivalences, while hK is a weak equivalence only if K is flag.
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There is a similar result in algebraic mc. The algebraic analogue of the lo op functor is the cobar construction : dgc dga. Thm 11. There is a htpy commutative diagram
ho colimdgc C (v )K -K ho colimdga [u]K -



K



,

(Q[K ]) -K - colimdga [u]K in which K and K are weak equivalences, while K is a weak equivalence only if K is flag. Cor 12. H( DJ (K ); Q) = H ho colimdga [u]K ( ) H ho colimdgl CL(u)K . ( DJ (K )) Z Q =
( )



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Ex 13. Let K b e the 1-skeleton of a 3-simplex. A calculation using the previous results gives H( DJ (K )) T (u1, u2, u3, u4, w123, w124, w134, w123) , = (relations) where deg wij k = 4 and there are 3 typ es of relations: (a) exterior algebra relations for u1, u2, u3, u4; (b) [ui, wj kl ] = 0 for i {j, k, l}; (c) [u1, w234]+[u2, w134]+[u3, w124]+[u4, w123] = 0. wij k is the higher commutator (Hurevicz image of the higher Samelson pro duct) of ui, uj and uk , so the last equation is a higher Jacobian identity.

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[1] Victor M Buchstab er and Taras E Panov. Torus Actions and Their Applications in Top ology and Combinatorics. Volume 24 of University Lecture Series, Amer. Math. So c., Providence, R.I., 2002.

[2] . . , . . . (in Russian), , , 2004. (Extended version of [1]).

[3] Michael W Davis and Tadeusz Januszkiewicz, Convex p olytop es, Coxeter orbifolds and torus actions, Duke Math. J. 62(2):417-451, 1991.

[4] Taras Panov, Nigel Ray and Rainer Vogt. Colimits, Stanley-Reiner algebras, and lo op spaces, in: "Categorical Decomp osition Techniques in Algebraic Top ology" (G.Arone et al eds.), Progress in Mathematics, vol. 215, BirkhЕ auser, Basel, 2004, pp. 261-291; arXiv:math.AT/0202081.
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