Higher Whitehead products in moment-angle complexes
Semyon Abramyan
NRU Higher School of Economics

Conference “Toric Topology 2019 in Okayama”
Okayama, 18–22 November 2019

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Whitehead products

Fix [fi ] ∈ πpi (X ), i = 1, . . . m and set p = p1 + . . . pm .

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Whitehead products

Fix [fi ] ∈ πpi (X ), i = 1, . . . m and set p = p1 + . . . pm .
The higher Whitehead product [f1 , . . . , fm ] ⊆ πp (X ) is a set of homotopy
classes of maps
(S)∂∆m⊆

S p−1

FS p1 ∨ · · · ∨ S pm

(S) mX
f1 ∨···∨fmpt.

Here the upper left arrow is the attaching map of p-cell in the productm(S)∆ = S p1 × · · · × S pm .

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Whitehead products in moment-angle complexes
We will consider iterated higher Whitehead products of maps:i-th inclusionµi : S 2 = CP 1 ⊂ CP ∞ −−−−−−−→ (CP ∞ )K .

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Whitehead products in moment-angle complexes
We will consider iterated higher Whitehead products of maps:i-th inclusionµi : S 2 = CP 1 ⊂ CP ∞ −−−−−−−→ (CP ∞ )K .

Theorem (Buchstaber-Panov)
The moment-angle complex (D 2 , S 1 )K is the homotopy fibre of thecanonical inclusion (CP ∞ )K ,→ (CP ∞ )m .

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

3/7Whitehead products in moment-angle complexes
We will consider iterated higher Whitehead products of maps:i-th inclusionµi : S 2 = CP 1 ⊂ CP ∞ −−−−−−−→ (CP ∞ )K .

Theorem (Buchstaber-Panov)
The moment-angle complex (D 2 , S 1 )K is the homotopy fibre of thecanonical inclusion (CP ∞ )K ,→ (CP ∞ )m .
Therefore, any iterated higher Whitehead product lifts to a map(D 2 , S 1 )K(CP ∞ )K(CP ∞ )mw

S d(w ) .
Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 20193/7

Main problem
There are classes of simplicial complexes for which corresponding
moment-angle complexes are homotopy equivalent to a wedge of spheres,for example, shifted (Grbić-Theriault), totally fillable (Iriye-Kishimoto).
Moreover, in these cases every sphere maps to (CP ∞ )K by (iteratedhigher) Whitehead product.
This leads us to the follwing problem.

Problem (Buchstaber-Panov)Assume the ZK is homotopy equivalent to a wedge of spheres. Is it truethat all wedge summands are represented by lifts S P → ZK of iterated
higher Whitehead products of the canonical maps µi : S 2 → (CP ∞ )K ?
From the discussion above follows that for shifted, totally fillable and, inaddition, flag complexes (Grbić-Panov-Theriault-Wu) the answer for theproblem is positive.
Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 20194/7

An example of unrealizability

Theorem (A.)
Let K be the simplicial complex(∂∆2 (1, 2, 3) ∗ ∂∆2 (4, 5, 6)) ∪ ∆2 (1, 2, 3) ∪ ∆2 (4, 5, 6).
ThenZK ' (S 7 )∨6 ∨ (S 8 )∨6 ∨ (S 9 )∨2 ∨ S 10 ,but S 10 cannot be realized by any linear combination of iterated higher
Whitehead products.

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Sketch of proof

Let X be a (desired) wedge (S 7 )∨6 ∨ (S 8 )∨6 ∨ (S 9 )∨2 ∨ S 10 . By
'Bahri-Bendersky-Cohen-Gitler theorem, we have spliting f : ΣX −→ ΣZK .
And because of dimensional reasons this equivalence can be desuspended
to f 0 : X → ZK . Thus, ZK is a desired wedge.
The only Whitehead products of dimension 10 are of the form[µi1 , [µi2 , µi3 , µi4 , µi5 , µi6 ]],[µi1 , µi2 , [µi3 , µi4 , µi5 , µi6 ]].

Both types are not defined because of combinatorics of K.

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 2019

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Positive answer

But still class of simplicial complexes with positive answer is large enough.

Theorem (A.)
Let K be a simplicial complex from the class, i.e. ZK is wedge of spheresand each sphere is a lift of Whitehead product. Then for any n ∈ Z>0 thecomplex (K ∗ ∂∆n ) ∪ ∆n is from the class too.

Semyon Abramyan (NRU HSE)

Higher Whitehead products

Okayama, November 22, 20197/7