ls category of quaternionic grassmannians · morse theory 1 ls category ls category and critical...

LS category of quaternionic Grassmannians

MarıaJose Pereira-SaezJoint work with E. Macıas-Virgos (Santiago de Compostela) and D. Tanre (Lille 1)

Lisboa, February 2015

M.J. Pereira-Saez (UDC) LS category of Gk,n Lisboa, 2015 1 / 28

LS category

1 LS categoryLS category and critical points

2 Morse theoryHeight functions on Sp(n)Height functions on the Grassmannian Gn,k

Cartan Model

3 Explicit coverings


LS category

Definition

The LS category cat(X ) is the least integer m ≥ 0 such that X admits acovering by m + 1 open sets which are contractible in X .

Given X a topological space, we say that the open set U ⊂ X iscategorical if U is contractible in X .

– For example, cat(Rn) = 0.

– All the spheres Sn, n ≥ 1 have LS category 1 (two contractible opensets)

– In general, all suspensions can be covered by two categorical opensets (cat=1).


LS category

Bounds

If X is a CW-complex, catX ≤ dim X .

If X is (r − 1)-connected for r ≥ 1, then

catX ≤ dim X/r .

The cup-length of a topological space X is a lower bound for thecategory of the space X :

cupX ≤ catX .

Where the cup-length of a space X , cup(X ), is the largest integer ` such that

there exists a product x1 · · · x` 6= 0, with xi ∈ H∗(X ;A), for any coefficient ring A.


LS category

Example: Quaternionic Grassmannians

With these bounds we can get the value of the LS category for themanifold Gk,n = Sp(n)/(Sp(k)× Sp(n − k)):

– Gn,k is 3-connected and its dimension is 4k(n − k) hence

catGk,n ≤4k(n − k)

3 + 1= k(n − k)

– The cohomology ring with integer coefficients of Gn,k is

H∗(Gn,k ;Z) ∼= (Z[q1, . . . , qk ]⊗Z[q1, . . . , qn−k ])/( ∑i+j=l

qi ⊗qj ; l ≥ 1),

where qi is the Pontrjagin class of degree 4i . Then, due to thenon-nullity of the class (q1 · · · qk)n−k we have

catGk,n ≥ k(n − k)


LS category LS category and critical points



Cartan Model



LS category LS category and critical points

If CritX denotes the minimum number of critical points for anysmooth function on a compact manifold X then we may write

catX + 1 ≤ CritX .

In the case of a finite number of critical points, Y. Rudyak andF. Schlenk have given a better upper bound.

Theorem (Rudyak-Schlenk, 2003)

Let X be a compact connected manifold and f : X → R any smoothfunction with a finite number of critical points. Then catX + 1 is a lowerbound for the number of critical values of f .


Morse theory



Cartan Model



Morse theory Height functions on Sp(n)



Cartan Model



Morse theory Height functions on Sp(n)

The symplectic group Sp(n) = A ∈ Hn×n | AA∗ = I is embedded inthe space Hn×n of quaternionic matrices of order n.

Let us consider the Morse height function hD : Sp(n)→ R withrespect to the hyperplane orthogonal to a real diagonal matrixD = diag(1, 2, . . . , n), defined by

hD(A) = <Tr(DA)

Its gradient is given by

(grad hD)A =1

2(D − ADA),

and its critical set is

Σ(hD) = diag(ε1, . . . , εn) with εi = ±1.


Morse theory Height functions on the Grassmannian Gn,k



Cartan Model




Let us fix some integer k ≤ n and consider the orbit

F = Fk = BJkB∗ | B ∈ Sp(n) ⊂ Sp(n)

of J = Jk = diag(−Ik , In−k).

F is diffeomorphic to the Grassmannian Gn,k of k-planes in Hn.

That is, we are considering the embedding Gn,k ⊂ Sp(n) which sendsthe k-dimensional subspace V onto the linear map which is equal to−id on V and to +id on V⊥.



Let hFD : F → R be the restriction of the height function hD to F .The relationship between the critical sets of hD and hFD is as follows.

Proposition

Σ(hFD ) = Σ(hD) ∩ F .



Recall that the critical points of hD are the matricesA = diag(ε1, . . . , εn) with εi = ±1.

If A has also to be in F , then A has exactly k negative elements andn − k positive elements.

So there are(nk

)critical points of hD in F .



Proposition

The Morse function, hFD : Fk∼= Gn,k → R, has exactly k(n− k) + 1 critical

levels.

So,

cupGn,k = k(n − k) ≤ catGn,k ≤ k(n − k).



For a given A ∈ Σ(hFD ),

hFD (A) = <Tr(DA) = 1ε1 + 2ε2 + · · ·+ nεn.

So, the maximum value of hFD at a critical point is

M = (−1− · · · − k) + (k + 1 + · · ·+ n).

And the minimum is

m = 1 + · · ·+ (n − k)− (n − k + 1)− · · · − n.

Hence, the number of critical levels is k(n − k) + 1 (as desired).


Morse theory Cartan Model



Cartan Model




Let us modify the embedding of the Grassmann manifold Gn,k intoSp(n) by using its structure of symmetric space.

The subgroup Sp(k)× Sp(n − k) is the fixed point set of theautomorphism σ : Sp(n)→ Sp(n) defined by σ(B) = JBJ.

Definition

The Cartan embedding γ : Gn,k∼= Sp(n)/Sp(k)× Sp(n − k)→ Sp(n) is

given by γ([B]) = Bσ(B)∗ = BJB∗J and the Cartan model M is theimage of γ. This image is the connected component of the identity of

N = B ∈ Sp(n) | σ(B) = B∗.

MJ equals the orbit F ⊂ Sp(n) of J by the conjugation action.



As Morse function on M, we choose the restriction to M of the heightfunction hD : Sp(n)→ R. We denote it by hM

D : M → R.

Proposition

The critical set of the Morse function, hMD : M ∼= Gn,k → R, is formed by

the matrices

A = diag(ε1, . . . , εk , εk+1, . . . , εn), εi = ±1,

such that the numbers of −1’s, in the blocks diag(ε1, . . . , εk) anddiag(εk+1, . . . , εn), are the same.

In general, these functions have too many critical points and levels(e.g.: 8 critical values in G5,3, while catG5,3 = 6).



The quaternionic projective space

When k = 1, the projective space Gn,1 = HPn−1 is diffeomorphic tothe orbit F = F1 of the diagonal matrix J = diag(−1, In−1) ∈ Sp(n).

We apply the two methods in this situation.



First, we consider the height function hFD . Its critical points are

Σ(hFD ) = A1, . . . ,An,

withAi = diag(1, . . . , 1,−1(i), 1, . . . , 1), for 1 ≤ i ≤ n.

The corresponding set of critical values is

n(n + 1)/2− 2i | 1 ≤ i ≤ n.



Second, let hMD be the Morse function induced on the Cartan model.

Its critical set is given by

Σ(hMD ) = E1,E2, . . . ,En

with E1 = I and

Ei = diag(−1, 1, . . . , 1,−1(i), 1, . . . , 1), for 2 ≤ i ≤ n.

Then, each critical point has a different image so, we have(n

1

)= n

critical values.

That is, for projective spaces, the sets of critical values obtained bythe two methods have the same cardinal, n, and cat HPn−1 = n − 1.


Explicit coverings



Cartan Model



Explicit coverings

We can build an explicit covering of the Cartan model M, bycontractible open sets obtained from a generalized Cayley transform.

Since MJ = F , this covering can be transformed in order to obtain acovering of the orbit F .


Explicit coverings

– Set Ω(J) = A ∈ Hn×n | A + J is invertible.

Theorem (Macıas-Virgos, P-S, Tanre)

The map cJ : Ω(J)→ Ω(J) given by

cJ(A) = (I − JA)(A + J)−1

verifies cJ cJ = id and induces a diffeomorphism,

cJ : TJF∼=−→ Ω(J) ∩ F .

– By translating ΩF (J) := Ω(J) ∩ F we obtain a finite covering of theorbit F = Fk

∼= Gn,k , associated to the critical set of hFD :


Explicit coverings

Let P = Pσ ∈ Rn×n be any real matrix obtained from the identitymatrix by a permutation σ ∈ Sn of rows. Any critical point of hFD isof the form AP = PJP∗.

We define

ΩF (AP) := PΩF (J)P∗ = A ∈ F | A + AP is invertible.

Proposition

The family (ΩF (APσ))σ∈Sn is a finite covering of the orbit F ∼= Gn,k bycontractible open sets.


Explicit coverings

Notice that the number of open sets in the covering of Gn,k equalsthat of critical points (the permutations of the matrix J), that is,

(nk

).

This number is in general strictly greater than the category of Gn,k ,except if k = 1 (quaternionic projective space).


Explicit coverings

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Lisboa, February 2015


ls category of quaternionic grassmannians · morse theory 1 ls category ls category and critical...

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