Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2013khabarovsk-talk.pdf Äàòà èçìåíåíèÿ: Thu Sep 5 09:07:42 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:13:58 2016 Êîäèðîâêà:

Homotopy theory of moment-angle complexes
sed on joint work with telen qriD tephen heriult nd tie u rs nov
Lomonosov Moscow State University

orus etionsX opologyD qeometry nd xumer heory uhrovskD P!U eptemer PHIQ

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

1 / 15


1. Problems

identifying the homotopy type of the momentEngle omplex ZK for ertin simpliil omplexes KY desriing the multiplition nd higher wssey produts in the TorElger H (ZK ) = Tork[v1 ,...,vm ] (k[K], k) of the fe ring k[K]Y desriing the oned lger reltionsY

Extk[

K]

(k, k) in terms of genertors nd

desriing the struture of the ontrygin lger H (DJ (K)) nd its ommuttor sulger H (ZK ) vi iterted nd higher hitehed @melsonA produtsY identifying the homotopy type of the loop spes DJ (K) nd ZK F

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

2 / 15


2. Preliminaries
(X , A) pir of spesF K simpliil omplex on [m] = {I, P, . . . , m}D
qiven I = {i1 , . . . , ik } [m]D set

KF

(X , A)I = Y1 â · · · â Ym

where Yi =

X A

if i I , if i I . /

he KEpolyhedrl produt of (X , A) is

(X , A)K =
I K

(X , A)I =
I K i I

Xâ
i I /

A.

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

3 / 15


Example
1

(X , A) = (D 2 , S 1 )D ZK := (D 2 , S 1 )K the momentEngle omplexF st hs n tion of the torus T m F (X , A) = (CP , pt )D DJ (K) := (CP , pt )K the hvis!tnuszkiewiz speF (X , A) = (C, Câ )D U (K) := (C, Câ )K = Cm
{i1 ,...,ik }K /

2

3

{zi1 = · · · = zik = H}

the omplement of oordinte suspe rrngementF

Proposition
here exists deformtion retrtion

Z
Taras Panov (MSU)

K

U (K) - Z

K
Khabarovsk 5 Sep 2013 4 / 15

Homotopy theory of m-a complexes


here exists homotopy (rtion

Z

K

-

D J (K ) (CP , pt )K

-

(CP )

m

(D 2 , S 1 )K
whih splits fter loopingX

(CP , CP )K

DJ (K)

ZK â T

m

rningX this is not n H Espe splitting

Proposition
here exists n ext sequene of nonommuttive lgers I - H ( ZK ) - H ( DJ (K)) - [u1 , . . . , um ] - I where [u1 , . . . , um ] denotes the exterior lger nd deg ui = IF
Taras Panov (MSU) Homotopy theory of m-a complexes Khabarovsk 5 Sep 2013 5 / 15

Ab


vet k denote Z or (eldF he fe ring @the tnley!eisner ringA of K is given y

k[K] := k[v1 , . . . , vm ] vi1 · · · vik = H if {i1 , . . . , ik } K /
where deg vi = PF

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

k is (eld
dui = vi , dvi = H KI = K|
I

H

-|I |-1

(KI )

k[K] is qolod ring if the multiplition nd ll higher wssey opertions in Tork[v1 ,...,vm ] (k[K], k) re trivilF
Taras Panov (MSU) Homotopy theory of m-a complexes Khabarovsk 5 Sep 2013 6 / 15


3. The case of a ag complex
e missing fe of K is suset I [m] suh tht I KD ut every proper / suset of I is simplex of KF

K is )g omplex if eh of its missing fes hs two vertiesF
iquivlentlyD K is )g if ny set of verties of K whih re pirwise onneted y edges spns simplexF

{)g omplexes on [m]} {simple grphs on [m]} K K() K
1

1-1

@oneEskeletonA



where K() is the lique omplex of @(ll in eh lique of with simplexAF
Taras Panov (MSU) Homotopy theory of m-a complexes Khabarovsk 5 Sep 2013 7 / 15


e grph is hordl if eh of its yles with

R edges hs hordF RF

iquivlentlyD is hordl if there re no indued yles of length

Theorem (FulkersonGross)
e grph is hordl if nd only if its verties n e ordered in suh wy thtD for eh vertex i D the lesser neighours of i form liqueF @perfet elimintion orderingA

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

8 / 15


Theorem (Grbic-P-Theriault-Wu) K is )gD k (eldF he following re equivlentX
1 2 3 4

k[K] is qolod ringY
the multiplition in H (ZK ) is trivilY

= K1 is hordl grphY ZK hs homotopy type of wedge of spheresF

¤ he equivlene (I) (P) (Q) ws proved y ferglund nd tollenek in PHHUF smplitions (I) (P)D (P) (Q) nd (R) (I) re vlid for ritrry KF roweverD (Q) @RA fils in the nonE)g seF

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

9 / 15


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

he nonzero ohomology groups of ZK re

H 0 = Z,

H 5 = Z10 ,

H 6 = Z15 ,

H 7 = Z6

H 9 = Z/P.
ell wssey produts vnish for dimensionl resonsD so K is qolod @over ny (eldAF

xeverthelessD ZK is not homotopy equivlent to wedge of spheres euse of the torsionF sn ftD

Z

K

(S 5 )

10

(S 6 )

15

(S 7 )6 7 RP 2 .

Question
essume tht H (ZK ) hs trivil multiplitionD so tht K is qolodD over ny (eldF ss it true tht ZK is oEH EspeD or even suspensionD s in the previous exmplec
Taras Panov (MSU) Homotopy theory of m-a complexes Khabarovsk 5 Sep 2013 10 / 15


row mny spheres in the wedgec ht if ZK is not wedge of spheresc

Theorem
por ny )g omplex KD there is n isomorphism

H DJ (K) T u1 , . . . , u =

m

ui2 = H, ui uj + uj ui = H for {i , j } K

where T u1 , . . . , um is the free lger on m genertors of degree IF ememer the ext sequene of nonEommuttive lgers I - H ( ZK ) - H ( DJ (K)) - [u1 , . . . , um ] - I

Ab

Proposition
por ny )g omplex KD the oinr¡ series of H (ZK ) is given y e I , (I - h1 t + · · · + (-I)n hn t n )

P H (ZK ); t

=

(I + t )m

-n

where h(K) = (h0 , h1 . . . , hn ) is the hEvetor of KF
Taras Panov (MSU) Homotopy theory of m-a complexes Khabarovsk 5 Sep 2013 11 / 15


Theorem
essume tht K is )gF he lger H (ZK )D viewed s the ommuttor sulger of H (DJ (K))D is multiplitively generted y 0 I [m] dim H (KI ) iterted ommuttors of the form

[uj , ui ],

[uk1 , [uj , ui ]],

...,

[uk1 , [uk2 , · · · [ukm-2 , [uj , ui ]] · · · ]]

where k1 < k2 < · · · < kp < j > i D ks = i for ny s D nd i is the smllest vertex in onneted omponent not ontining j of the suomplex K{k1 ,...,kp ,j ,i } F purthermoreD this multiplitive generting set is minimlF

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

12 / 15


rere is n importnt prtiulr se @orresponding to K = m pointsAF st is n nlogue of the desription of sis in the ommuttor sulger of free lgerD given y gohen nd xeisendorferX

Lemma
vet A e the ommuttor sulger of T u1 , . . . , um /(ui2 = H)X I - A - T u1 , . . . , um /(ui2 = H) - [u1 , . . . , um ] - I where deg ui = IF hen A is free ssoitive lgerF st is minimlly generted y the iterted ommuttors of the form

[uj , ui ],

[uk1 , [uj , ui ]],

...,

[uk1 , [uk2 , · · · [ukm-2 , [uj , ui ]] · · · ]]
is equl to ( - I)
m

where k1 < k2 < · · · < kp < j > i nd ks = i for ny s F he numer of ommuttors of length
Taras Panov (MSU)

F
13 / 15

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013


Corollary
essume tht K is )g nd ZK hs homotopy type of wedge of spheresF hen the numer of spheres of dimension + I in the wedge is given y 0 mF |I |= dim H (KI )D for P sn prtiulrD H i (KI ) = H for i > H nd ll I F

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

14 / 15


Reference

telen qriD rs novD tephen heriult nd tie uF romotopy types of momentEngle omplexesF reprint @PHIPAY rivXIPIIFHVUQF

Taras Panov (MSU)

Homotopy theory of m-a complexes

Khabarovsk 5 Sep 2013

15 / 15