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MSC[2020] Primary: 55M30, 57S15; Secondary: 15B33, 57T10

Keywords: Lusternik-Schnirelmann category; Lie group; Quasi-projective space; Quaternionic logarithm

Introduction↩︎

The problem list of T. Ganea ([1]) still contains unresolved questions on the determination of Lusternik-Schnirelmann category (in short, LS-category). In particular, the LS-category of classical Lie groups is still incomplete. The complex case was completely solved by W. Singhof ([2], [3]) with \(\mathop{\mathrm{cat}}{\mathrm{SU}}(n)=n-1\). In the quaternionic case, the LS-category of the symplectic group, \({\mathrm{Sp}}(n)\), is known only in low dimensions: \(\mathop{\mathrm{cat}}\,{\mathrm{Sp}}(1)=\mathop{\mathrm{cat}}\,S^3=1\), \(\mathop{\mathrm{cat}}\,{\mathrm{Sp}}(2)=3\) ([4]) and \(\mathop{\mathrm{cat}}\,{\mathrm{Sp}}(3)=5\) ([5]). For the higher cases, there are few informations. One knows from [6] that \(\mathop{\mathrm{cat}}\,{\mathrm{Sp}}(n)\geq n+2\) if \(n\geq 3\), and from [7] that \(\mathop{\mathrm{cat}}\,{\mathrm{Sp}}(n)\leq n(n+1)/2\).

In this work, we are interested in quasi-projective spaces. In the complex case, they coincide with the suspension of the projective spaces \(\mathbb{C}{\mathrm{P}}^n\) but, in the quaternionic case, the situation is different. Their cellular structure still determines the cohomology ring of the group \({\mathrm{Sp}}(n)\) ([8]) but they are not the third suspension of the quaternionic projective spaces \(\mathbb{H}{\mathrm{P}}^n\). Denote them by \(Q_{n}\) and look at the first cases.

For \(n=1\), we have the sphere \(S^3\). For \(n=2\) we get a 2-cell space \(Q_{2}=S^3\cup_{\varphi}e^7\) where \(\varphi\) is a generator of \(\pi_{6}(S^3)\). It has a non-zero Hopf invariant, thus \(\mathop{\mathrm{cat}}Q_{2}=2\), see [9]. The first difficult case, \(n=3\), was solved by N. Iwase and T. Miyauchi ([10]) who proved \(\mathrm{cat}\,Q_{3}=3\).

In contrast with this result, let us notice that [11] implies that the subspace \(Q_{3}\) has subspace category 2 in \({\mathrm{Sp}}(3)\). The proof relies on the knowledge of some homotopy groups of \(Q_{2}\), \({\mathrm{Sp}}(2)\), \({\mathrm{Sp}}(3)\). We prove here that this is a general fact and the proof can be done “à la Singhof” from the description of \(Q_{n}\) and \({\mathrm{Sp}}(n)\) as sets of particular matrices and the use of a quaternionic logarithm.

Let us observe that the product cells of \({\mathrm{SU}}(n)\) are grafted onto the quasi-projective complex space of LS-category 1. In the case of \({\mathrm{Sp}}(n)\), the quasi-projective quaternionic space is not a suspension. Moreover it is not even a space of constant LS-category for any \(n\), as shown by the cases \(n=2\) and \(n=3\). But, for a determination of the LS-category of \({\mathrm{Sp}}(n)\), the main point is not the LS-category of the quasi-projective space but its relative LS-category in \({\mathrm{Sp}}(n)\). Our result shows that the two situations are similar. It reinforces the intuition of the existence of an upper bound for the LS-category of \({\mathrm{Sp}}(n)\) in the form of a polynomial in \(n\) of degree 1, the most challenging conjecture being \(\mathop{\mathrm{cat}}{\mathrm{Sp}}(n)\leq 2n-1\).

The basic definitions—such as those of quasi-projective spaces, subspace LS-category, quaternionic logarithm, and so on—are recalled in the first section. The second section contains the proof of the Main theorem.

1 Background↩︎

Let us begin with the symplectic group, \({\mathrm{Sp}}(n)\). We denote by \(\mathbb{H}\) the (non-commutative) field of quaternions and \(\mathbb{H}^n\) the \(n\)-dimensional vector space over \(\mathbb{H}\), with a right \(\mathbb{H}\)-action. If \(x=(x_{i})_{1\leq i\leq n}\in \mathbb{H}^n\) and \(y=(y_{i})_{1\leq i\leq n}\in \mathbb{H}^n\), we define the scalar product \(\langle x,y\rangle=\sum_{i=1}^n\overline{x}_{i}y_{i}\), where \(\overline{x}_{i}\) is the conjugate of \(x_{i}\in\mathbb{H}\). The group \({\mathrm{Sp}}(n)\) is the group of orthogonal transformations of \(\mathbb{H}^n\) or, equivalently, of the quaternionic matrices of order \(n\) such that \(A^*A=I_{n}\). We embedd \(\mathbb{H}^n\) in \(\mathbb{H}^{n+1}\) by putting the last coordinate equal to zero, and \({\mathrm{Sp}}(n)\) in \({\mathrm{Sp}}(n+1)\) by \[A\mapsto \begin{pmatrix} A&0\cr 0&1\cr \end{pmatrix}.\]

The use of the logarithm in the study of LS-category of Lie groups takes its origin in the determination \(\mathop{\mathrm{cat}}{\mathrm{SU}}(n)=n-1\) made by Singhof in [2], [3]. Let us recall the definition of a principal branch of the logarithm for quaternions.

The Lie group of unit quaternions \(S^3\) has for Lie algebra the set \(\mathbb{H}_0=\langle\mathbf{i},\mathbf{j},\mathbf{k}\rangle\) of skew-symmetric quaternions \(q\); i.e., those of the form \(t\,\mathbf{\omega}\) for some imaginary unit quaternion \(\mathbf{\omega}\) and some real number \(t\in\mathbb{R}\). Any quaternion \(q\in \mathbb{H}\) can be written \[q=s+t \,\mathbf{\omega}, \quad s,t\in\mathbb{R}, t\geq 0, \quad s^2+t^2=\vert q \vert^2.\] By taking the argument \(\theta\in [0,+\pi)\), we get the polar form \[q= \vert q \vert e^{\theta\,\mathbf{\omega}}=\vert q \vert\left(\cos\theta + (\sin\theta)\, \mathbf{\omega}\right).\] The principal branch of the quaternionic logarithm of \(q=s+t\mathbf{\omega}\) is then defined ([12]) as \[{\mathrm{Log}\,}q = \ln \vert q \vert + \theta\,\mathbf{\omega}=\ln\vert q \vert + \arccos \Re\left(\frac{q}{\vert q \vert}\right)\,\mathbf{\omega},\] where \(\theta\in [0,+\pi)\) and \(\Re\) is the real part. This function is continuous except for \(\theta=\pi\), which corresponds to the quaternions \(q=-\vert q \vert\) on the negative real axis. Let us notice that two similar quaternions (i.e. same norm, same real part) have similar logarithms.

Let \(S^{4n-1}\) be the unit sphere of \(\mathbb{H}^n\). The quaternionic quasi-projective space, \(Q_{n}\), is the image of the map \[\varphi\colon S^{4n-1}\times S^3\to {\mathrm{Sp}}(n),\] defined by \[\varphi(x,\lambda)=x(\lambda -1)x^*+I_{n}.\] (More geometrically, the map \(\varphi(x,\lambda)\) sends \(x\) to \(x\lambda\) and is the identity on the subspace orthogonal to \(x\).) Equivalently, \(Q_{n}\) is the quotient space of \(S^{4n-1}\times S^3\) by the equivalence relation \[(x,\lambda)\sim (x\nu,\nu^{-1}\lambda\nu), \text{ with } \nu\in S^3, \text{ and } (x,1)\sim (y,1)\text{ for any } x,\,y\in S^{4n-1}.\] The group law \(\mu\) of \({\mathrm{Sp}}(n)\) induces a relative homeomorphism ([8]) \[\mu\colon (Q_{n}\times {\mathrm{Sp}}(n-1),Q_{n-1}\times {\mathrm{Sp}}(n-1)) \to ({\mathrm{Sp}}(n),{\mathrm{Sp}}(n-1)),\] which maps \(Q_{n}\times {\mathrm{Sp}}(n-1)\) onto \({\mathrm{Sp}}(n)\).

Let \(E^{4(n-1)}\) be the ball consisting of all vectors \(x\in S^{4n-1}\subset \mathbb{H}^n\) with \(x_{n}\in \mathbb{R}\) and \(x_{n}\geq 0\). We denote \(h_{n}\) the composition of \(\varphi\) with the product of the inclusion map \(\iota_{n}\colon E^{4(n-1)}\to S^{4n-1}\) and the canonical surjection \(\rho\colon E^3\to S^3\cong E^3/S^2\), \[h_{n}\colon E^{4n-1}=E^{4(n-1)}\times E^3\xrightarrow{\iota_{n}\times \rho} S^{4n-1}\times S^3 \xrightarrow{\varphi}Q_{n}.\] The characteristic maps of the cells of \({\mathrm{Sp}}(n)\) are products of maps \(h_{i}\), \[E^{4i_{1}-1}\times\cdots\times E^{4i_{r}-1} \xrightarrow{h_{i_{1}}\times\cdots\times h_{i_{r}}} Q_{i_{1}}\times\cdots\times Q_{i_{r}} \xrightarrow{\mu} {\mathrm{Sp}}(n),\] where \(n\geq i_{1}>i_{2}>\dots >i_{r}>0\). Such a cell is denoted by \((i_{1},\dots,i_{r})\) and called a normal cell. In [8], Steenrod shows that \({\mathrm{Sp}}(n)\) is a CW-complex whose cells are the normal cells and the 0-cell \(I_{n}\).

If \(X\) is a metric ANR, one gets the same integer if the \(U_{i}\)’s are any subsets of \(X\), see [13]. It is true for instance for any CW-complex.

2 Proof of the Main Theorem↩︎

We need three subsets whose union is \(Q_{n}\), each one being contractible in \({\mathrm{Sp}}(n)\). We set:

• \(O_{1}=\left\{\varphi(x,\lambda)\mid x\in S^{4n-1},\,\lambda\neq 1,\,\lambda\neq -1\right\}\),

• \(O_{2}=\left\{\varphi(x,-1)\mid x\in S^{4n-1}\right\}\) and

• \(O_{3}\) an open ball containing \(I_{n}\).

By definition, they cover \(Q_{n}\) and the open set \(O_{3}\) is contractible. We now use the quaternionic logarithm to obtain an explicit formula for the contracting homotopies in \(O_{1}\) and \(O_{2}\).

Let us begin with \(A=\varphi(x,\lambda)=x(\lambda -1)x^*+I_{n}\in O_{1}\). We choose an orthonormal basis \((x,Y)\) of \(\mathbb{H}^n\), with \(Y=(y_{2},\dots,y_{n})\in \mathbb{H}^{n-1}\), such that \[A=\left(x \;Y\right) \begin{pmatrix}\lambda&0\\ 0&I_{n-1} \end{pmatrix} \addtolength{\arraycolsep}{-3pt}\begin{pmatrix}x^*\cr Y^*\end{pmatrix}.\] As \(\lambda\neq -1\) the quaternionic logarithm is defined by \({\mathrm{Log}\,}A = x ({\mathrm{Log}\,}\lambda) x^*\) and we have a continuous map, from \(O_{1}\) to \({\mathrm{Sp}}(n)\), \[{\mathrm{Log}\,}A = \left(x \;Y\right) \begin{pmatrix} {\mathrm{Log}\,}\lambda&0\\ 0&I_{n-1} \end{pmatrix} \addtolength{\arraycolsep}{-3pt}\begin{pmatrix}x^*\cr Y^*\end{pmatrix}.\] By using the exponential, we get \[\begin{align} A=e^{{\mathrm{Log}\,}A} & = & e^{x({\mathrm{Log}\,}\lambda)x^*} \\ &=& 1+\sum_{n=1}^{\infty} \frac{1}{n!} (x({\mathrm{Log}\,}\lambda)x^*)^n = 1+x\left(\sum_{n=1}^{\infty} \frac{1}{n!} ({\mathrm{Log}\,}\lambda)^n\right)x^*\\ &=& 1+x\left(e^{{\mathrm{Log}\,}\lambda}-1\right)x^*=1+x(\lambda -1)x^*=\varphi(x,\lambda). \end{align}\] We define a continuous map from \(O_{1}\times [0,1]\) to \({\mathrm{Sp}}(n)\) by \[\Phi(A,t)=e^{x ((1-t){\mathrm{Log}\,}\lambda) x^*}.\] The extremities of this path are \(\Phi(A,0)=e^{{\mathrm{Log}\,}A}=A\) and \(\Phi(A,1)=I_{n}\). A computation similar to the previous one also gives \[\Phi(A,t)=1+x\left(e^{(1-t){\mathrm{Log}\,}\lambda}-1\right)x^*=\varphi(x,e^{(1-t){\mathrm{Log}\,}\lambda})\] and the contractibility of \(O_{1}\).

We continue with \(A=\varphi(x,-1)\in O_{2}\). From the definition, we have \(\varphi(x,-1)=\varphi(x,\mathbf{i})\varphi(x,\mathbf{i})\). We define a continuous map \(\Psi\colon O_{2}\times [0,1]\to {\mathrm{Sp}}(n)\) by \[\Psi(A,t)=\left\{ \begin{array}{ccl} \varphi(x,e^{(1-2t){\mathrm{Log}\,}\mathbf{i}})\varphi(x,\mathbf{i})&\quad\text{if}&0\leq t\leq 1/2,\\ \varphi(x,e^{2(1-t){\mathrm{Log}\,}\mathbf{i}})&\quad\text{if}&1/2\leq t\leq 1. \end{array}\right.\] We observe that \(\Psi(A,0)=\varphi(x,\mathbf{i})\varphi(x,\mathbf{i})=\varphi(x,-1)\) and \(\Psi(A,1)=\varphi(x,1)=I_{n}\).

In the previous proof, the subsets \(O_{1}\) and \(O_{3}\) are contractible and not only contractible in \({\mathrm{Sp}}(n)\). In contrast, this is not the case for \(O_{2}\): this open subset is contractible in \({\mathrm{Sp}}(n)\) but the contraction does not stay in \(Q_{n}\). This is coherent with the fact that the LS-category of \(Q_{n}\) is greater than 3 ([10]) for \(n\geq 3\).

Enrique Macías-Virgós
CITMAga, Departamento de Matemáticas,
Universidade de Santiago de Compostela,
15782 Santiago de Compostela, Spain

Daniel Tanré
Département de Mathématiques,
UMR-CNRS 8524, Université de Lille,
59655 Villeneuve d’Ascq Cedex, France

References↩︎