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Rational asp ects of toric top ology
Taras Panov Moscow State University Joint work with Nigel Ray (Manchester )

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1. Motivations and definitions. Object of study: T n-manifolds M 2n and their orbit quotients. Notation: T := T n, M := M 2n, Q := M /T . Particular examples:

· Non-singular compact toric varieties
T -action is a part of algebraic C n -action with a dense orbit;

· (Quasi)toric manifolds of Davis-Januszkiewicz
"lo cally standard" (i.e., lo cally lo ok like T n acting on Cn ) and Q combinatorially is a simple p olytop e;

· Torus manifolds of Hattori-Masuda.

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K a simplicial complex on V = {v1, . . . , vm} (e.g., K is the dual to the b oundary of Q). S (V ) a symmetric algebra on V over a ring R, deg vi = 2. Given V , set v := i vi. The Stanley-Reisner algebra (or face ring ) of K is R[K ] := S (V )/(v : K ). / The Davis-Januszkiewicz space DJ (K ) := Prop erties: · DJ (K ) E T вT M for K = ( Q); · H (DJ (K ); R) = HT (M ; R) = R[K ]. Define
) m. ZK := hofibre DJ (K ) B T
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K

B T B T m = (CP )m.

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Thus, we have two homotopy pullback diagrams ZK


- ET m


M 2n and


- E n T


DJ (K ) - B T m

DJ (K ) - B T n

The map DJ (K ) B T n is determined by a choice of a regular sequence in the Cohen-Macaulay algebra Z[K ] = H (DJ (K )). Overal aim: Relate

· Top ology of M , ZK , DJ (K ) and their lo op spaces;

· Combinatorics of Q, K ; · Commutative and homological algebra of Q[K ]

through rational homotopy theory
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2. Rational homotopy theory. Sullivan's framework: Piecewise p olynomial differential forms functor A : top cdga together with a natural isomorphism
= (X )) - H (X, Q) H (A

for any X top. The algebra A(X ) may b e thought of as a commutative replacement for the singular Q-co chains. A space X is formal if A(X ) is a formal dga, i.e. if there is a weak equivalence A(X ) H (X ). In particular, X is formal if there is a multiplicative "choice of a representative" map H (X, Q) A(X ).

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Quillen's framework: Quillen's approach is dual (in the Eckmann-Hilton sense) to Sullivan's. The rational homotopy groups (X ) Z Q of a space X form a graded Lie algebra, called the rational homotopy Lie algebra of X , with resp ect to the Samelson bracket. There is a Quillen functor Q : top1 dgl0 from p ointed simply connected spaces to connected differential graded Lie algebras, with a natural isomorphism H (Q(X )) - (X ) Z Q for any X top1. A space X is called coformal if Q(X ) is a coformal differential graded Lie algebra. In particular, X is coformal if there is a weak equivalence Q(X ) (X ) Z Q.
=

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3. Cohomology of DJ (K ), ZK and M . According to a result of Notb ohm-Ray, there is a commutative diagram H (B T m) = S (V ) -


A(B T m)


H (DJ (K )) = Q[K ] - A(DJ (K )) in which the horizontal arrows are weak equivalences. In particular, DJ (K ) is formal. Applying A( ) to the pullback square defining ZK , we get A(ZK )


A(E T m) -


.

A(DJ (K )) A(B T m) - This may fail to b e a pushout square in cdga.

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Nevertheless, an Eilenb erg-Mo ore typ e result implies that the induced map
) (DJ (K )), A (B T m ), A (E T m ) A (Z ) BA K (

is a quism. Consider the free extension diagram (U ) Q[K ] (U ) S (V ) -


,

Q[K ] - S (V ) where (U ) = [u1, . . . , um], deg ui = 1, and the dga structure in (U ) Q[K ] and (U ) S (V ) is defined by dvi = 0, dui = vi. Thm 1 The free extension (U ) Q[K ] of the Stanley-Reisner ring Q[K ] is weakly equivalent to A(ZK ). Hence, there are isomorphisms of (bi)graded algebras H (ZK ; Q) = TorS (V )(Q[K ], Q) = H ((U ) Q[K ]). The statement is also true with Z co efficients, although the pro of uses different techniques. It has also b een proven by M. Franz in his thesis in a slightly different context of non-compact nonsingular toric varieties.
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The following is a generalisation of an argument due to Bousfield-Gugenheim. Prop 2 Let B b e a simply connected space and t1, . . . , tn a sequence of elements in H 2(B ; Z). Then we have a pullback diagram of fibre bundles E - E n T
.

B - BT n Assume that B is formal and H (B ; Q) is free as a S (t1, . . . , tn)-mo dule. Then E is also formal and H (E ; Q) = H (B ; Q)/(t1, . . . , tn). Cor 3 All toric manifolds are formal. The same argument works also in a more general case of torus manifolds over homology p olytop es. Even more generally, torus manifolds M with H odd(M ) = 0 are also formal. In this case the face ring Z[K ] has to b e replaced by the face ring Z[S ] of an appropriate simplicial p oset S determined by M . This face ring still admits an lsop t1, . . . tn and is free as a Z[t1, . . . , tn]-mo dule.
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4. Flag complexes and lo op spaces 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 (V )
/(

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

vivj = 0 for {i, j } K , / a quadratic algebra. The fibration DJ (K ) B T m with fibre ZK splits after lo oping: DJ (K ) ZK в T m. This is not an H -space, and the exact sequence of Pontrjagin homology rings 0 H(ZK ) H( DJ (K )) (U ) 0 do es not split in general.

)

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Thm 4 Supp ose K is flag and k a field. Then H( DJ (K )) = Tk(U )
/(

u2 = 0, i

uiuj + uj ui = 0 for {i, j } K . Idea of pro of. Use Adams' cobar construction on the S-R coalgebra k[K ]: H( DJ (K )) = H (C(DJ (K ))) = Extk[K ](k, k) Then use a result of FrЁ erg calculating the latter ob Ext. Cor 5 We have an isomorphism of graded Lie algebras: ( 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.

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The Poincarґ series of R[K ] is given by e F (R[K ]; t) = fit2(i+1) (1 - t2)i+1 i=-1 h0 + h1t2 + . . . + hnt2n = , (1 - t2)n
n -1

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. Cor 6 For any flag complex K the rational homology Poincarґ series of DJ (K ) is given by e (1 + t)n F H( DJ (K )); t = . nhn tn 1 - h1t + . . . + (-1) Pro of. Since H( DJ (K ); Q) is the quadratic dual of the Stanley-Reisner algebra Q[K ] (graded by deg vi = 1), we have F Q[K ]; -t · F H( DJ (K )); t = 1.
( ) ( ) ( )

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5. Diagrams and homotopy colimits. cat(K ): category of K (simplices and inclusions); mc: a mo del category (e.g., top 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 ] = colimdgc C (v )K , etc., where C (v ) is the symmetric coalgebra on v , deg v = 2. Cor 7 Assume K is flag. Then H( DJ (K ), Q) = colimga [u]K ; ( DJ (K )) Z Q = colimgl CL(u)K , where CL(u) is the commutative Lie algebra, deg u = 1.
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In general colimit mo dels do not work! (Lo ok at K = 2.) Thm 8 DJ (K ) is coformal iff K is flag. Pro of. For flag K construct a map C(DJ (K )) H( DJ (K ); Q) and show that it is a quism using the ab ove homology calculation. For non-flag K higher Samelson and commutator brackets app ear in and H, obstructing coformality.

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Thm 9 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 10
( )



H( DJ (K ); Q) = H ho colimdga [u]K ( ) H ho colimdgl CL(u)K . ( DJ (K )) Z Q = Compare: Thm 11 (P.-Ray-Vogt) There is a htpy commutative diagram
h ho colimtop B T K - K ho colimtgrp T K -

p

K



,

DJ (K )

-K -

h

colimtgrp 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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