Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2015beijing-talk.pdf Äàòà èçìåíåíèÿ: Fri Nov 27 16:46:10 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:07:51 2016 Êîäèðîâêà:

Cohomology of quotients of moment-angle manifolds.
Taras Panov
Moscow State University

International Open Chinese-Russian Conference Torus Actions: Topology, Geometry and Number Theory School of Mathematics and Systems Science Chinese Academy of Sciences Beijing, 26­29 October 2015

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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1. Basics

K a simplicial complex on [m] = {1, . . . , m}. For each simplex I = {i1 , . . . , ik } K, set (D 2 , S 1 )I = {(x1 , . . . , xm ) (D 2 )m : xi S 1 = D 2 when i I }. / The moment-angle complex is the polyhedral product ZK = (D 2 , S 1 )K =
I K

(D 2 , S 1 )I (D 2 )m .

ZK is a manifold whenever K is a triangulated sphere, and can be smoothed when K is a boundary of a polytope or is a starshaped sphere (comes from a complete simplicial fan).

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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Also define BT
K

= (CP , pt )K =
K

I K

BT I BT

m

= (CP )m .

The cohomology of BT the face ring of K:

(with coefficients in a commutative ring R ) is

H (BT K ) R [K] = R [v1 , . . . , vm ]/(vi1 · · · vik : {i1 , . . . , ik } K), / = where deg vi = 2. There is a homotopy fibration ZK - BT
K

- BT

m

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Quotients of m-a manifolds

26­29 Oct 2015

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2. Partial quotients
The torus T m acts on ZK coordinatewise. We consider freely acting subtori H T m and partial quotients ZK /H . The manifolds ZK /H have recently attracted attention as they support complex-analytic structures, usually non-K¨ ahler, with interesting geometry. We turn the face ring R [K] into a module over the polynomial ring H (B (T m /H )) via the homomorphism H (B (T m /H )) H (BT m ) = R [v1 , . . . , vm ] R [K]

Theorem
For any commutative ring R , there is an isomorphism of graded algebras H (ZK /H ; R ) TorH =


(B (T m /H );R )

(R [K], R ).

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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3. Proof of the main theorem
The Eilenberg­Moore spectral sequence of the homotopy fibration ZK /H BT K B (T m /H ) has E2 = TorH and converges to H (ZK /H ) TorC =


(B (T m /H ))

(R [K], R )

(B (T m /H ))

(C (BT K ), R ).

We shall establish a multiplicative isomorphism TorH


(B (T m /H ))

(R [K], R ) TorC



(B (T m /H ))

(C (BT K ), R );

it would also imply the collapse of the Eilenberg­Moore spectral sequence.
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 5 / 15


For any torus T k we consider the map of R -modules : H (BT k ) = (H (BT 1 ))k - (C (BT 1 ))k - C (BT k ), where C denotes the normalised singular cochain functor with coefficients in R , the map i is the k -fold tensor product of the map H (BT 1 ) = R [v ] C (BT 1 ) sending v to any representing cochain, and â is the k -fold cross-product. The map induces an isomorphism in cohomology. We have R [K] = H (BT K ) = lim H (BT I )
I K i â

where each H (BT I ) is a polynomial ring on |I | generators, the (inverse) limit is taken in the category of graded algebras for the diagram consisting of projections H (BT I ) H (BT J ) corresponding to J I K.
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 6 / 15


Now consider the diagram H (BT I ) =R [K]=H (BT K ) limI K C (BT I ) (B (T m /H )) - (colim I K R - C C I K BT ) = C (BT ) lim
I K

R - H (B (T m /H )) - R - C (B (T m /H )) -

where the double arrows denote derivatives of and the horizontal arrows on the right are induced by the maps BT I BT m BT m /H . All vertical arrows above induce isomorphisms in cohomology (for the bottom right arrow this follows from excision). If the diagram was commutative in the category da of differential graded algebras (i.e. consisted of multiplicative maps), then the standard functoriality of Tor would have implied the required isomorphism TorH


(B (T m /H ))

(R [K], R ) TorC =



(B (T m /H ))

(C (BT K ), R ) H (ZK /H ). =
26­29 Oct 2015 7 / 15

Taras Panov (Moscow University)

Quotients of m-a manifolds


H (BT I ) =R [K]=H (BT K ) limI K C (BT I ) (B (T m /H )) - (colim I K R - C C I K BT ) = C (BT ) lim
I K

R - H (B (T m /H )) - (B (T m /H )) - R - C

The lower part of the diagram is indeed a commutative diagram in da. The upper part is not commutative though, and the double arrow maps are not morphisms in da as is not multiplicative. Nevertheless, Tor enjoys extended functoriality with respect to morphisms in the category dash, provided that the diagram above is homotopy commutative in dash, by [Munkholm74, 5.4]. The objects of dash are the same as in da, while morphisms A A are coalgebra maps BA BA of the bar constructions. The map and the double arrows above are morphisms in dash by [Munkholm74, 7.3].
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 8 / 15


H (BT I ) =R [K]=H (BT K ) limI K C (BT I ) (B (T m /H )) - (colim I K R - C C I K BT ) = C (BT ) lim
I K

R - H (B (T m /H )) - R - C (B (T m /H )) -

To see that the upper right square is homotopy commutative, it is enough to establish the homotopy commutativity of the diagram H (B (T m /H )) - H (BT I ) - H (BT J ) (B (T m /H )) - (BT I ) - (BT J ) C C C for any J I K. The right square is commutative in the standard sense by the construction of (note that we are using normalised cochains), while the left square is homotopy commutative by [Munkholm74, 7.3].
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 9 / 15


It remains to prove that the isomorphism TorH


(B (T m /H ))

(R [K], R ) TorC



(B (T m /H ))

(C (BT K ), R )

is multiplicative. Let us take a closer look on how the product structure is defined on both sides.

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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We have a commutative diagram
R R R - - C (B (T m /H )) C (B (T m /H )) - C (B (T m /H )) - C (BT K ) C (BT K ) C (BT K )

Using the functoriality of Tor in dash we get a natural map TorC


(B (T m /H ))C (B (T m /H ))

(C (BT K ) C (BT K ), R R )
/H ))

TorC (B (T m which, composed with the classical Kunneth-like map ¨ TorC


(C (BT K ), R )

(B (T m /H ))


(C (BT K ), R ) TorC



TorC

(B (T m /H ))C (B (T m /H ))

(C (BT ) C (BT K ), R R ),


(B (T m /H )) K

(C (BT K ), R ) (C (BT K ), R ).

gives the multiplicative structure in TorC

(B (T m /H ))

It can be checked that this multiplicative structure is the same as the one defined via the Eilenberg­Zilber theorem and used in the Eilenberg­Moore isomorphism TorC (B (T m /H )) (C (BT K ), R ) H (ZK /H ). =
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 11 / 15


The product in TorH



(B (T m /H ))

(H (BT K ), R ) is defined similarly.

Denote B = C (B (T m /H )) and M = C (BT K ). The diagram

TorHB (HM , R ) TorHB (HM , R ) - TorHB HB (HM HM , R R ) - TorB (M , R ) TorB (M , R ) - TorB B (M M , R R ) - in which the vertical arrows are isomorphisms of R -modules, is commutative, because the corresponding 3-dimensional diagram in which each TorB (M , R ) is replaced by R B M is homotopy commutative in dash. Therefore, the R -module isomorphism TorHB (HM , R ) TorB (M , R ) is multiplicative with respect to the multiplicative structure given. The proof is complete.
Taras Panov (Moscow University) Quotients of m-a manifolds 26­29 Oct 2015 12 / 15


4. Remarks and examples
When R is a field of zero characteristic, one can avoid appealing to the category dash by using a commutative cochain model in the argument above. One can also avoid using dash when H is a trivial subgroup, as in [Buchstaber-Panov15, Ex. 8.1.12]. Examples of quotients ZK /H include compact toric manifolds M (when H has maximal possible dimension). In this case R [K] is a free H (B (T m /H ))-module, and Theorem 1 reduces to the well-known description of the cohomology: H (M ; R ) TorH =


(B (T m /H );R )

(R [K], R ) = R [K] p H (B (T m /H )).

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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Another series 1 ZK /Sd corresp When K is the structure as an

of examples are `projective' moment-angle manifolds 1 onding to the diagonal subcircle H = Sd T m . 1 admits a complex-analytic boundary of a polytope, ZK /Sd LVM-manifold.

In this case Theorem 1 together with the Koszul resolution gives the following isomorphism:
1 H (ZK /Sd ) H ([t1 , . . . , t = m -1

] R [K ], d )

where the cohomology of the differential graded algebra on the right hand side is taken with respect to dti = vi - vm , dvj = 0, deg ti = 1. There is also a similar description of the Dolbeault cohomology of the complex quotients ZK /H [Panov-Ustinovsky12].

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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References
V. Buchstaber, T. Panov. Toric Topology. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2015, 518 pages. H. Munkholm. The Eilenberg­Moore spectral sequence and strongly homotopy multiplicative maps. J. Pure Appl. Algebra 5 (1974), 1­50. T. Panov, Yu. Ustinovsky. Complex-analytic structures on moment-angle manifolds. Moscow Math. J. 12 (2012), no. 1, 149­172. T. Panov. On the cohomology of quotients of moment-angle complexes. Preprint (2015) arXiv:1506.06875.

Taras Panov (Moscow University)

Quotients of m-a manifolds

26­29 Oct 2015

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