Editing Projective linear group (section)

== Topology ==
Over the real and complex numbers, the topology of PGL and PSL can be determined from the [[fiber bundle]]s that define them:
: <math>\begin{matrix}
\mathrm{ Z} &\cong& K^\times &\to& \mathrm{GL} &\to& \mathrm{PGL} \\
\mathrm{SZ} &\cong& \mu_n &\to& \mathrm{SL} &\to& \mathrm{PSL}
\end{matrix}</math>
via the [[long exact sequence of a fibration]].

For both the reals and complexes, SL is a [[covering space]] of PSL, with number of sheets equal to the number of ''n''th roots in ''K''; thus in particular all their higher [[homotopy groups]] agree. For the reals, SL is a 2-fold cover of PSL for ''n'' even, and is a 1-fold cover for ''n'' odd, i.e., an isomorphism:
: {{nowrap|{{mset|±1}} → SL(2''n'', '''R''') → PSL(2''n'', '''R''')}}
: {{nowrap|SL(2''n'' + 1, '''R''') {{overset|lh=0.6|~|→}} PSL(2''n'' + 1, '''R''')}}
For the complexes, SL is an ''n''-fold cover of PSL.

For PGL, for the reals, the fiber is {{nowrap|'''R'''<sup>×</sup> ≅ {{mset|±1}}}}, so up to homotopy, {{nowrap|GL → PGL}} is a 2-fold covering space, and all higher homotopy groups agree.

For PGL over the complexes, the fiber is {{nowrap|'''C'''<sup>×</sup> ≅ ''S''<sup>1</sup>}}, so up to homotopy, {{nowrap|GL → PGL}} is a [[circle bundle]]. The higher homotopy groups of the circle vanish, so the homotopy groups of {{nowrap|GL(''n'', '''C''')}} and {{nowrap|PGL(''n'', '''C''')}} agree for {{nowrap|''n'' ≥ 3}}. In fact, ''π''<sub>2</sub> always vanishes for Lie groups, so the homotopy groups agree for {{nowrap|''n'' ≥ 2}}. For {{nowrap|1=''n'' = 1}}, we have that {{nowrap|1=''π''<sub>1</sub>(GL(''n'', '''C''')) = ''π''<sub>1</sub>(''S''<sup>1</sup>) = '''Z'''}}. The fundamental group of {{nowrap|PGL(2, '''C''')}} is a finite cyclic group of order 2.

=== Covering groups ===
Over the real and complex numbers, the projective special linear groups are the ''minimal'' ([[centerless]]) [[Lie group]] realizations for the special linear Lie algebra <math>\mathfrak{sl}(n)\colon</math> every connected Lie group whose Lie algebra is <math>\mathfrak{sl}(n)</math> is a cover of {{nowrap|PSL(''n'', ''F'')}}. Conversely, its [[universal covering group]] is the ''maximal'' ([[simply connected]]) element, and the intermediary realizations form a [[Covering group#Lattice of covering groups|lattice of covering groups]].

For example, [[SL2(R)|{{nowrap|SL(2, '''R''')}}]] has center {{mset|±1}} and fundamental group '''Z''', and thus has universal cover {{nowrap|{{overline|SL(2, '''R''')}}}} and covers the centerless {{nowrap|PSL(2, '''R''')}}.