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'''Burnside's lemma''', sometimes also called '''Burnside's counting theorem''', the '''Cauchy–Frobenius lemma''' or the '''orbit-counting theorem''', is a result in [[group theory]] which is often useful in taking account of [[symmetry]] when counting mathematical objects. Its various eponyms include [[William Burnside]], [[George Pólya]], [[Augustin Louis Cauchy]], and [[Ferdinand Georg Frobenius]]. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to {{harvtxt|Frobenius|1887}}.<ref>{{harvnb|Burnside|1897|loc=§119}}</ref>
[[File:Covering space diagram.svg|thumb|250px|right|A covering map satisfies the local triviality condition. Intuitively, such maps locally project a "stack of pancakes" above an [[open set|open region]], ''U'', onto ''U''.]]
In the following, let ''G'' be a [[finite set|finite]] [[group (mathematics)|group]] that [[Group action|acts]] on a [[Set (mathematics)|set]] ''X''. For each ''g'' in ''G'' let ''X<sup>g</sup>'' denote the set of [[element (mathematics)|element]]s in ''X'' that are [[fixed point (mathematics)|fixed by]] ''g''. Burnside's lemma asserts the following formula for the number of [[orbit (group theory)|orbit]]s, denoted |''X''/''G''|:<ref>{{harvnb|Rotman|1995|loc=Chapter 3}}</ref>
In [[mathematics]], more specifically [[algebraic topology]], a '''covering map''' (also '''covering projection''') is a [[continuous function|continuous]] [[Function (mathematics)|function]] ''p'' <ref>{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |year=1966 |publisher=McGraw-Hill |page=62|isbn=}}</ref> from a [[topological space]], ''C'', to a topological space, ''X'', such that each point in ''X'' has an [[open set|open]] neighbourhood '''evenly covered''' by ''p'' (as shown in the image); the precise definition is given below. In this case, ''C'' is called a '''covering space''' and ''X'' the '''base space''' of the covering projection. The definition implies that every covering map is a [[local homeomorphism]].
Covering spaces play an important role in [[homotopy theory]], [[harmonic analysis]], [[Riemannian geometry]] and [[differential topology]]. In Riemannian geometry for example, [[Ramification#In_algebraic_topology|ramification]] is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the [[fundamental group]]. An important application comes from the result that, if ''X'' is a "sufficiently good" [[topological space]], there is a [[bijection]] from the collection of all isomorphism classes of [[connected space|connected coverings]] of ''X'' and subgroups of the [[fundamental group]] of ''X''.
Thus the number of orbits (a [[natural number]] or [[Extended real number line|+∞]]) is equal to the [[mean|average]] number of points fixed by an element of ''G'' (which is also a natural number or infinity). If ''G'' is infinite, the division by |''G''| may not be well-defined; in this case the following statement in [[cardinal arithmetic]] holds:
== Formal definition ==
:<math>|G| |X/G| = \sum_{g \in G}|X^g|.</math>
Let ''X'' be a [[topological space]]. A '''covering space''' of ''X'' is a space ''C'' together with a [[continuous function (topology)|continuous]] [[surjective]] map
:<math>p \colon C \to X\,</math>
== Example application ==
The number of rotationally distinct colourings of the faces of a [[Cube (geometry)|cube]] using three colours can be determined from this formula as follows.
such that for every {{nowrap|''x'' ∈ ''X''}}, there exists an [[open set|open]] [[neighborhood (topology)|neighborhood]] ''U'' of ''x'', such that ''p''<sup>−1</sup>(''U'') (the [[inverse image]] of ''U'' under ''p'') is a union of disjoint open sets in ''C'', each of which is mapped [[homeomorphism|homeomorphically]] onto ''U'' by ''p''.<ref name="Chernavskii">{{harvnb|Chernavskii|2001}}</ref><ref name="Munkres p336">{{harvnb|Munkres|2000|p=336}}</ref>
Let ''X'' be the set of 3<sup>6</sup> possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group ''G'' of the cube act on ''X'' in the natural manner. Then two elements of ''X'' belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the [[fixed set]]s for the 24 elements of ''G''.
The map ''p'' is called the '''covering map''',<ref name="Munkres p336"/> the space ''X'' is often called the '''base space''' of the covering, and the space ''C'' is called the '''total space''' of the covering. For any point ''x'' in the base the inverse image of ''x'' in ''C'' is necessarily a [[discrete space]]<ref name="Munkres p336"/> called the [[Fiber (mathematics)|fiber]] over ''x''.
[[Image:Face colored cube.png|thumb|Cube with coloured faces]]
* one identity element which leaves all 3<sup>6</sup> elements of ''X'' unchanged
* six 90-degree face rotations, each of which leaves 3<sup>3</sup> of the elements of ''X'' unchanged
* three 180-degree face rotations, each of which leaves 3<sup>4</sup> of the elements of ''X'' unchanged
* eight 120-degree vertex rotations, each of which leaves 3<sup>2</sup> of the elements of ''X'' unchanged
* six 180-degree edge rotations, each of which leaves 3<sup>3</sup> of the elements of ''X'' unchanged
The special open neighborhoods ''U'' of ''x'' given in the definition are called '''evenly-covered neighborhoods'''. The evenly-covered neighborhoods form an [[open cover]] of the space ''X''. The homeomorphic copies in ''C'' of an evenly-covered neighborhood ''U'' are called the '''sheets''' over ''U''. One generally pictures ''C'' as "hovering above" ''X'', with ''p'' mapping "downwards", the sheets over ''U'' being horizontally stacked above each other and above ''U'', and the fiber over ''x'' consisting of those points of ''C'' that lie "vertically above" ''x''. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism, ''h'', from the pre-image ''p''<sup>−1</sup>(''U''), of an evenly covered neighbourhood ''U'', onto {{nowrap|''U'' × ''F''}}, where ''F'' is the fiber, satisfying the '''local trivialization condition''', which is that, if we project {{nowrap|''U'' × ''F''}} onto ''U'', {{nowrap|''π'' : ''U'' × ''F'' → ''U''}}, so the composition of the projection ''π'' with the homeomorphism ''h'' will be a map {{nowrap|''π'' ∘ ''h''}} from the pre-image ''p''<sup>−1</sup>(''U'') onto ''U'', then the derived composition {{nowrap|''π'' ∘ ''h''}} will equal ''p'' locally (within ''p''<sup>−1</sup>(''U'')).
A detailed examination of these automorphisms may be found
[[Cycle index#The cycle index of the face permutations of a cube|here]].
===Alternative definitions===
The average fix size is thus
Many authors impose some [[connectedness|connectivity]] conditions on the spaces ''X'' and ''C'' in the definition of a covering map. In particular, many authors require both spaces to be [[path-connected]] and [[locally path-connected]].<ref>{{Cite book|title = An Introduction to Knot Theory|date = 1997|last = Lickorish|pages = 66-67}}</ref><ref>{{Cite book|title = Topology and Geometry|last = Bredon|year = 1997|isbn = 978-0387979267}}</ref> This can prove helpful because many theorems hold only if the spaces in question have these properties. Some authors omit the assumption of surjectivity, for if ''X'' is connected and ''C'' is nonempty then surjectivity of the covering map actually follows from the other axioms.
* A topological space ''X'' has a [[universal cover]] if and only if it is connected, locally path connected, and semi-locally simply connected.
Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in ''n'' colors is given by
* <math>\mathbb{R}</math> is the universal cover of the unit circle ''S<sup>1</sup>''
* The [[spin group]] ''Spin(n)'' is a double cover of the [[special orthogonal group]] and a universal cover when ''n > 2''. The accidental, or [[exceptional isomorphism|exceptional isomorphisms]] for Lie groups then give isomorphisms between spin groups in low dimension and classical Lie groups.
== Proof ==
* The [[unitary group]] ''U(n)'' has universal cover ''SU(n) x R''
The first step in the proof of the lemma is to re-express the sum over the group elements ''g'' ∈ ''G'' as an equivalent sum over the set elements ''x'' ∈ ''X'':
* The [[n-sphere]] ''S<sup>n</sup>'' is a double cover of real projective space ''RP<sup>n</sup>'' and is a universal cover for ''n >1''.
* Every manifold has an [[orientable double cover]] that is connected if and only if the manifold is non-orientable.
(Here ''X<sup>g</sup>'' = {''x'' ∈ ''X'' | ''g.x'' = ''x''} is the subset of all points of ''X'' fixed by ''g'' ∈ ''G'', whereas ''G<sub>x</sub>'' = {''g'' ∈ ''G'' | ''g.x'' = ''x''} is the [[Orbit-stabilizer theorem#Orbits and stabilizers|stabilizer subgroup]] of G that fixes the point ''x'' ∈ ''X''.)
* The [[uniformization theorem]] asserts that every compact Riemann surface has a universal cover conformally equivalent to the [[Riemann sphere]], the complex plane, or the unit disc.
The [[Orbit-stabilizer theorem#Orbits and stabilizers|orbit-stabilizer theorem]] says that there is a natural [[bijection]] for each ''x'' ∈ ''X'' between the orbit of ''x'', ''G.x'' = {''g.x'' | ''g'' ∈ ''G''} ⊆ ''X'', and the set of left cosets ''G/G<sub>x</sub>'' of its stabilizer subgroup ''G<sub>x</sub>''. With [[Lagrange's theorem (group theory)|Lagrange's theorem]] this implies
* The universal cover of a wedge of ''n'' circles is the [[Cayley graph]] of the free group on ''n'' generators, i.e. a [[Bethe lattice]].
:<math>|G.x| = [G\,:\,G_x] = |G| / |G_x|.</math>
* The [[torus]] is a double cover of the [[Klein bottle]]
Our sum over the set ''X'' may therefore be rewritten as
* Every graph has a [[bipartite double cover]]. Since every graph is homotopic to a wedge of circles, its universal cover is a Cayley graph.
* Every immersion from a compact manifold to a manifold of the same dimension is a covering of its image.
Finally, notice that ''X'' is the disjoint union of all its orbits in ''X/G'', which means the sum over ''X'' may be broken up into separate sums over each individual orbit.
Putting everything together gives the desired result:
* Every cover {{nowrap|''p'': ''C'' → ''X''}} is a [[local homeomorphism]]<ref>{{harvnb|Munkres|2000|p=338}}</ref>—that is, for every {{nowrap|''c'' ∈ ''C''}}, there exists a neighborhood {{nowrap|''U'' ⊆ ''C''}} of ''c'' and a neighborhood {{nowrap|''V'' ⊆ ''X''}} of ''p''(''c'') such that the restriction of ''p'' to ''U'' yields a [[homeomorphism]] from ''U'' to ''V''. This implies that ''C'' and ''X'' share all local properties. If ''X'' is [[simply connected]] and ''C'' is connected, then this holds globally as well, and the covering ''p'' is a homeomorphism.
* If {{nowrap|''p'' : ''E'' → ''B''}} and {{nowrap|''p''′ : ''E''′ → ''B''′}} are covering maps, then so is the map {{nowrap|''p'' × ''p''′ : ''E'' × ''E''′ → ''B'' × ''B′''}} given by {{nowrap|1=(''p'' × ''p''′)(''e'', ''e''′) = (''p''(''e''), ''p''′(''e''′))}}.<ref>{{harvnb|Munkres|2000|p=339|loc=Theorem 53.3}}</ref>
For every ''x'' in ''X'', the fiber over ''x'' is a [[discrete space|discrete]] subset of ''C''.<ref name="Munkres p336"/> On every [[connected space|connected component]] of ''X'', the fibers are homeomorphic.
If ''X'' is connected, there is a discrete space ''F'' such that for every ''x'' in ''X'' the fiber over ''x'' is [[homeomorphism|homeomorphic]] to ''F'' and, moreover, for every ''x'' in ''X'' there is a neighborhood ''U'' of ''x'' such that its full pre-image ''p''<sup>−1</sup>(''U'') is homeomorphic to {{nowrap|''U'' × ''F''}}. In particular, the [[cardinality]] of the fiber over ''x'' is equal to the cardinality of ''F'' and it is called the '''degree of the cover''' {{nowrap|''p'' : ''C'' → ''X''}}. Thus, if every fiber has ''n'' elements, we speak of an '''''n''-fold covering''' (for the case {{nowrap|1=''n'' = 1}}, the covering is trivial; when {{nowrap|1=''n'' = 2}}, the covering is a '''double cover'''; when {{nowrap|1=''n'' = 3}}, the covering is a '''triple cover''' and so on).
==History: the lemma that is not Burnside's==
[[William Burnside]] stated and proved this lemma, attributing it to {{harvnb|Frobenius|1887}} in his 1897 book on finite groups. But, even prior to Frobenius, the formula was known to [[Cauchy]] in 1845. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy. Consequently, this lemma is sometimes referred to as '''the lemma that is not Burnside's'''.<ref>{{harvnb|Neumann|1979}}</ref> (see also [[Stigler's law of eponymy]]). This is less ambiguous than it may seem: Burnside contributed many lemmas to this field.
===Lifting properties===
== See also ==
{{see also|Homotopy lifting property}}
* [[Pólya enumeration theorem]]
If {{nowrap|1=''p'' : ''C'' → ''X''}} is a cover and γ is a path in ''X'' (i.e. a continuous map from the [[unit interval]] {{nowrap|[0, 1]}} into ''X'') and {{nowrap|''c'' ∈ ''C''}} is a point "lying over" γ(0) (i.e. {{nowrap|1=''p''(''c'') = γ(0))}}, then there exists a unique path Γ in ''C'' lying over γ (i.e. {{nowrap|1=''p'' ∘ Γ = γ}}) such that with {{nowrap|1=Γ(0) = ''c''}}. The curve Γ is called the '''lift''' of γ. If ''x'' and ''y'' are two points in ''X'' connected by a path, then that path furnishes a [[bijection]] between the fiber over ''x'' and the fiber over ''y'' via the lifting property.
More generally, let {{nowrap|''f'' : ''Z'' → ''X''}} be a continuous map to ''X'' from a [[path connected]] and [[locally path connected]] space ''Z''. Fix a base-point {{nowrap|''z'' ∈ ''Z''}}, and choose a point {{nowrap|''c'' ∈ ''C''}} "lying over" ''f''(''z'') (i.e. {{nowrap|1=''p''(''c'') = ''f''(''z'')}}). Then there exists a '''lift''' of ''f'' (that is, a continuous map {{nowrap|''g'' : ''Z'' → ''C''}} for which {{nowrap|1=''p'' ∘ ''g'' = ''f''}} and {{nowrap|1=''g''(''z'') = ''c''}}) [[if and only if]] the [[induced homomorphism]]s {{nowrap|''f''<sub>#</sub> : π<sub>1</sub>(''Z'', ''z'') → π<sub>1</sub>(''X'', ''f''(''z''))}} and {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''f''(''z''))}} at the level of [[fundamental groups]] satisfy
Moreover, if such a lift ''g'' of ''f'' exists, it is unique.
In particular, if the space ''Z'' is assumed to be [[simply connected]] (so that {{nowrap|π<sub>1</sub>(''Z'', ''z'')}} is trivial), condition {{EquationNote|♠|(♠)}} is automatically satisfied, and every continuous map from ''Z'' to ''X'' can be lifted. Since the unit interval {{nowrap|[0, 1]}} is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.
If {{nowrap|''p'' : ''C'' → ''X''}} is a covering and {{nowrap|''c'' ∈ ''C''}} and {{nowrap|''x'' ∈ ''X''}} are such that {{nowrap|1=''p''(''c'') = ''x''}}, then ''p''<sub>#</sub> is injective at the level of [[fundamental groups]], and the induced homomorphisms {{nowrap|''p''<sub>#</sub> : π<sub>''n''</sub>(''C'', ''c'') → π<sub>''n''</sub>(''X'', ''x'')}} are [[group isomorphism|isomorphisms]] for all {{nowrap|''n'' ≥ 2}}. Both of these statements can be deduced from the lifting property for continuous maps. Surjectivity of ''p''<sub>#</sub> for {{nowrap|''n'' ≥ 2}} follows from the fact that for all such ''n'', the ''n''-sphere '''S'''<sup>''n''</sup> is simply connected and hence every continuous map from '''S'''<sup>''n''</sup> to ''X'' can be lifted to ''C''.
===Equivalence===
Let {{nowrap|''p''<sub>1</sub> : ''C''<sub>1</sub> → ''X''}} and {{nowrap|''p''<sub>2</sub> : ''C''<sub>2</sub> → ''X''}} be two coverings. One says that the two coverings ''p''<sub>1</sub> and ''p''<sub>2</sub> are '''equivalent''' if there exists a homeomorphism {{nowrap|''p''<sub>21</sub> : ''C''<sub>2</sub> → ''C''<sub>1</sub>}} and such that {{nowrap|1=''p''<sub>2</sub> = ''p''<sub>1</sub> ∘ ''p''<sub>21</sub>}}. Equivalence classes of coverings correspond to conjugacy classes of subgroups of the [[fundamental group]] of ''X'', as discussed below. If {{nowrap|''p''<sub>21</sub> : ''C''<sub>2</sub> → ''C''<sub>1</sub>}} is a covering (rather than a homeomorphism) and {{nowrap|1=''p''<sub>2</sub> = ''p''<sub>1</sub> ∘ ''p''<sub>21</sub>}}, then one says that ''p''<sub>2</sub> '''dominates''' ''p''<sub>1</sub>.
===Covering of a manifold===
Since coverings are local [[homeomorphism]]s, a covering of a topological ''n''-[[manifold]] is an ''n''-manifold. (One can prove that the covering space is [[second-countable]] from the fact that the [[fundamental group]] of a manifold is always [[countable]].) However a space covered by an ''n''-manifold may be a [[non-Hausdorff manifold]]. An example is given by letting ''C'' be the plane with the origin deleted and ''X'' the quotient space obtained by identifying every point {{nowrap|(''x'', ''y'')}} with {{nowrap|(2''x'', ''y''/2)}}. If {{nowrap|''p'' : ''C'' → ''X''}} is the quotient map then it is a covering since the action of ''Z'' on ''C'' generated by {{nowrap|1=''f''(''x'', ''y'') = (2''x'', ''y''/2)}} is [[properly discontinuous]]. The points {{nowrap|''p''(1, 0)}} and {{nowrap|''p''(0, 1)}} do not have disjoint neighborhoods in ''X''.
Any covering space of a differentiable manifold may be equipped with a (natural) differentiable structure that turns ''p'' (the covering map in question) into a [[local diffeomorphism]] – a map with constant [[Rank (differential topology)|rank]] ''n''.
== Universal covers ==
A covering space is a '''universal covering space''' if it is [[simply connected]]. The name ''[[universal property|universal]] cover'' comes from the following important property: if the mapping {{nowrap|''q'': ''D'' → ''X''}} is a universal cover of the space ''X'' and the mapping {{nowrap|''p'' : ''C'' → ''X''}} is any cover of the space ''X'' where the covering space ''C'' is connected, then there exists a covering map {{nowrap|''f'' : ''D'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f'' = ''q''}}. This can be phrased as
<BLOCKQUOTE >The universal cover of the space ''X'' covers all connected covers of the space ''X''.</BLOCKQUOTE >
The map ''f'' is unique in the following sense: if we fix a point ''x'' in the space ''X'' and a point ''d'' in the space ''D'' with {{nowrap|1=''q''(''d'') = ''x''}} and a point ''c'' in the space ''C'' with {{nowrap|1=''p''(''c'') = ''x''}}, then there exists a unique covering map {{nowrap|''f'' : ''D'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f''= ''q''}} and {{nowrap|1=''f''(''d'') = ''c''}}.
If the space ''X'' has a universal cover then that universal cover is essentially unique: if the mappings {{nowrap|''q''<sub>1</sub> : ''D''<sub>1</sub> → ''X''}} and {{nowrap|''q''<sub>2</sub> : ''D''<sub>2</sub> → ''X''}} are two universal covers of the space ''X'' then there exists a homeomorphism {{nowrap|''f'' : ''D''<sub>1</sub> → ''D''<sub>2</sub>}} such that {{nowrap|1=''q''<sub>2</sub> ∘ ''f'' = ''q''<sub>1</sub>}}.
The space ''X'' has a universal cover if it is [[Connected space|connected]], [[Connected space#Local connectedness|locally path-connected]] and [[semi-locally simply connected]]. The universal cover of the space ''X'' can be constructed as a certain space of paths in the space ''X''.
The example {{nowrap|'''R''' → '''S'''<sup>1</sup>}} given above is a universal cover. The map {{nowrap|'''S'''<sup>3</sup> → SO(3)}} from [[quaternion|unit quaternions]] to [[rotation]]s of 3D space described in [[quaternions and spatial rotation]] is also a universal cover.
If the space ''X'' carries some additional structure, then its universal cover usually inherits that structure:
* if the space ''X'' is a [[manifold]], then so is its universal cover ''D''
* if the space ''X'' is a [[Riemann surface]], then so is its universal cover ''D'', and ''p'' is a [[holomorphic]] map
* if the space ''X'' is a [[Lorentzian manifold]], then so is its universal cover. Furthermore, suppose the subset ''p''<sup>−1</sup>(''U'') is a [[disjoint union]] of open sets each of which is diffeomorphic with ''U'' by the mapping ''p''. If the space ''X'' contains a [[closed timelike curve]] (CTC), then the space ''X'' is [[timelike multiply connected]] (no CTC can be [[timelike homotopic]] to a point, as that point would not be causally well-behaved), its universal (diffeomorphic) cover is [[timelike simply connected]] (it does not contain a CTC).
* if ''X'' is a [[Lie group]] (as in the two examples above), then so is its universal cover ''D'', and the mapping ''p'' is a homomorphism of Lie groups. In this case the universal cover is also called the ''[[universal covering group]]''. This has particular application to [[representation theory]] and [[quantum mechanics]], since ordinary [[Group representation|representations]] of the universal covering group (''D'') are [[projective representation]]s of the original (classical) group (''X'').
The universal cover first arose in the theory of [[analytic functions]] as the natural domain of an [[analytic continuation]].
== G-coverings ==
Let ''G'' be a [[discrete group]] [[group action|acting]] on the [[topological space]] ''X''. It is natural to ask under what conditions the projection from ''X'' to the [[orbit space]] ''X''/''G'' is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product {{nowrap|''X'' × ''X''}} by the twist action where the non-identity element acts by {{nowrap|(''x'', ''y'') ↦ (''y'', ''x'')}}. Thus the study of the relation between the fundamental groups of ''X'' and ''X''/''G'' is not so straightforward.
However the group ''G'' does act on the fundamental groupoid of ''X'', and so the study is best handled by considering groups acting on groupoids, and the corresponding ''orbit groupoids''. The theory for this is set down in Chapter 11 of the book ''Topology and groupoids'' referred to below. The main result is that for discontinuous actions of a group ''G'' on a Hausdorff space ''X'' which admits a universal cover, then the fundamental groupoid of the orbit space ''X''/''G'' is isomorphic to the orbit groupoid of the fundamental groupoid of ''X'', i.e. the quotient of that groupoid by the action of the group ''G''. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.
A '''deck transformation''' or '''automorphism''' of a cover {{nowrap|''p'' : ''C'' → ''X''}} is a [[homeomorphism]] {{nowrap|''f'' : ''C'' → ''C''}} such that {{nowrap|1=''p'' ∘ ''f'' = ''p''}}. The set of all deck transformations of ''p'' forms a group under [[function composition|composition]], the '''deck transformation group''' Aut(''p''). Deck transformations are also called '''covering transformations'''. Every deck transformation [[permutation|permutes]] the elements of each fiber. This defines a [[group action]] of the deck transformation group on each fiber. Note that by the unique lifting property, if ''f'' is not the identity and ''C'' is path connected, then ''f'' has no [[fixed point (mathematics)|fixed points]].
Now suppose {{nowrap|''p'' : ''C'' → ''X''}} is a covering map and ''C'' (and therefore also ''X'') is connected and locally path connected. The action of Aut(''p'') on each fiber is [[group action|free]]. If this action is [[group action#Types of actions|transitive]] on some fiber, then it is transitive on all fibers, and we call the cover '''regular''' (or '''normal''' or '''Galois'''). Every such regular cover is a [[principal bundle|principal ''G''-bundle]], where {{nowrap|1=''G'' = Aut(''p'')}} is considered as a discrete topological group.
Every universal cover {{nowrap|''p'' : ''D'' → ''X''}} is regular, with deck transformation group being isomorphic to the [[fundamental group]] π<sub>1</sub>(''X'').
The example {{nowrap|''p'' : '''C'''<sup>×</sup> → '''C'''<sup>×</sup>}} with {{nowrap|1=''p''(''z'') = ''z''<sup>''n''</sup>}} from above is a regular cover. The deck transformations are multiplications with ''n''-th [[root of unity|roots of unity]] and the deck transformation group is therefore isomorphic to the [[cyclic group]] ''C''<sub>''n''</sub>.
Another example: {{nowrap|''p'' : '''C'''* → '''C'''*}} with {{nowrap|1=''p''(''z'') = ''z''<sup>''n''!</sup>}} from above is regular. Here one has a hierarchy of deck transformation groups. In fact ''C''<sub>''x''!</sub> is a subgroup of ''C''<sub>''y''!</sub>, for {{nowrap|1 ≤ ''x'' ≤ ''y'' ≤ ''n''}}.
== Monodromy action ==
{{main|Monodromy}}
Again suppose {{nowrap|''p'' : ''C'' → ''X''}} is a covering map and ''C'' (and therefore also ''X'') is connected and locally path connected. If ''x'' is in ''X'' and ''c'' belongs to the fiber over ''x'' (i.e. {{nowrap|1=''p''(''c'') = ''x''}}), and {{nowrap|γ : [0, 1] → ''X''}} is a path with {{nowrap|1=γ(0) = γ(1) = ''x''}}, then this path lifts to a unique path in ''C'' with starting point ''c''. The end point of this lifted path need not be ''c'', but it must lie in the fiber over ''x''. It turns out that this end point only depends on the class of γ in the fundamental group {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. In this fashion we obtain a right [[group action]] of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} on the fiber over ''x''. This is known as the '''monodromy action'''.
There are two actions on the fiber over {{nowrap|''x'' : Aut(''p'')}} acts on the left and {{nowrap|π<sub>1</sub>(''X'', ''x'')}} acts on the right. These two actions are compatible in the following sense:
<math>f\cdot(c\cdot\gamma) = (f\cdot c)\cdot\gamma</math> for all ''f'' in Aut(''p''), ''c'' in ''p''<sup>−1</sup>(''x'') and γ in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
If ''p'' is a universal cover, then Aut(''p'') can be naturally identified with the [[Dual (category theory)|opposite]] group of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} so that the left action of the opposite group of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} coincides with the action of Aut(''p'') on the fiber over ''x''. Note that Aut(''p'') and {{nowrap|π<sub>1</sub>(''X'', ''x'')}} are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through {{nowrap|''g'' ↦ ''g''<sup>−1</sup>)}}.
If ''p'' is a [[Covering_space#Deck_transformation_group.2C_regular_covers|regular]] cover, then Aut(''p'') is naturally isomorphic to a quotient of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
In general (for good spaces), Aut(''p'') is naturally isomorphic to the quotient of the [[Centralizer and normalizer|normalizer]] of {{nowrap|''p''<sub>*</sub>(π<sub>1</sub>(''C'', ''c''))}} in {{nowrap|π<sub>1</sub>(''X'', ''x'')}} over {{nowrap|''p''<sub>*</sub>(π<sub>1</sub>(''C'', ''c''))}}, where {{nowrap|1=''p''(''c'') = ''x''}}.
==More on the group structure==
Let {{nowrap|''p'' : ''C'' → ''X''}} be a covering map where both ''X'' and ''C'' are path-connected. Let {{nowrap|''x'' ∈ ''X''}} be a basepoint of ''X'' and let {{nowrap|''c'' ∈ ''C''}} be one of its pre-images in ''C'', that is {{nowrap|1=''p''(''c'') = ''x''}}. There is an [[induced homomorphism]] of [[fundamental group]]s {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'',''x'')}} which is injective by the lifting property of coverings. Specifically if ''γ'' is a closed loop at ''c'' such that {{nowrap|1=''p''<sub>#</sub>([''γ'']) = 1}}, that is {{nowrap|''p'' ∘ ''γ''}} is [[null-homotopic]] in ''X'', then consider a null-homotopy of {{nowrap|''p'' ∘ ''γ''}} as a map {{nowrap|''f'' : ''D''<sup>2</sup> → ''X''}} from the 2-disc ''D''<sup>2</sup> to ''X'' such that the restriction of ''f'' to the boundary '''S'''<sup>1</sup> of ''D''<sup>2</sup> is equal to {{nowrap|''p'' ∘ ''γ''}}. By the lifting property the map ''f'' lifts to a continuous map {{nowrap|''g'' : ''D''<sup>2</sup> → ''C''}} such that the restriction of ''f'' to the boundary '''S'''<sup>1</sup> of ''D''<sup>2</sup> is equal to ''γ''. Therefore ''γ'' is [[null-homotopic]] in ''C'', so that the [[Kernel (algebra)|kernel]] of {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''x'')}} is trivial and thus {{nowrap|''p''<sub>#</sub> : π<sub>1</sub>(''C'', ''c'') → π<sub>1</sub>(''X'', ''x'')}} is an injective homomorphism.
Therefore {{nowrap|π<sub>1</sub>(''C'', ''c'')}} is isomorphic to the subgroup {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. If {{nowrap|''c''<sub>1</sub> ∈ ''C''}} is another pre-image of ''x'' in ''C'' then the [[subgroup]]s {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} and {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''<sub>1</sub>))}} are [[conjugacy class|conjugate]] in {{nowrap|π<sub>1</sub>(''X'', ''x'')}} by ''p''-image of a curve in ''C'' connecting ''c'' to ''c''<sub>1</sub>. Thus a covering map {{nowrap|''p'' : ''C'' → ''X''}} defines a conjugacy class of subgroups of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} and one can show that equivalent covers of ''X'' define the same conjugacy class of subgroups of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
For a covering {{nowrap|''p'' : ''C'' → ''X''}} the group {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} can also be seen to be equal to
:<math>\Gamma_p(c) = \{ [\gamma] : \gamma_C \mbox{ is a closed curve in } C \mbox { passing through } c\in C \}</math>,
the set of [[homotopy class]]es of those closed curves γ based at ''x'' whose lifts ''γ<sub>C</sub>'' in ''C'', starting at ''c'', are closed curves at ''c''. If ''X'' and ''C'' are path-connected, the degree of the cover ''p'' (that is, the cardinality of any fiber of ''p'') is equal to the [[Index of a subgroup|index]] [{{nowrap|π<sub>1</sub>(''X'', ''x'') : ''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}}] of the [[subgroup]] {{nowrap|''p''<sub>#</sub>(π<sub>1</sub>(''C'', ''c''))}} in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}.
A key result of the covering space theory says that for a "sufficiently good" space ''X'' (namely, if ''X'' is path-connected, locally path-connected and [[semi-locally simply connected]]) there is in fact a bijection between equivalence classes of path-connected covers of ''X'' and the conjugacy classes of subgroups of the fundamental group {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. The main step in proving this result is establishing the existence of a universal cover, that is a cover corresponding to the trivial subgroup of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. Once the existence of a universal cover ''C'' of ''X'' is established, if ''H'' ≤ π<sub>1</sub>(''X'', ''x'') is an arbitrary subgroup, the space ''C''/''H'' is the covering of ''X'' corresponding to ''H''. One also needs to check that two covers of ''C'' corresponding to the same (conjugacy class of) subgroup of {{nowrap|π<sub>1</sub>(''X'', ''x'')}} are equivalent. Connected [[cell complex]]es and connected [[manifold]]s are examples of "sufficiently good" spaces.
Let ''N''(''Γ<sub>p</sub>'') be the [[normalizer]] of ''Γ<sub>p</sub>'' in {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. The deck transformation group Aut(''p'') is isomorphic to the [[quotient group]] ''N''(''Γ<sub>p</sub>'')/''Γ<sub>p</sub>''. If ''p'' is a universal covering, then ''Γ<sub>p</sub>'' is the [[trivial group]], and Aut(''p'') is isomorphic to π<sub>1</sub>(''X'').
Let us reverse this argument. Let ''N'' be a [[normal subgroup]] of {{nowrap|π<sub>1</sub>(''X'', ''x'')}}. By the above arguments, this defines a (regular) covering {{nowrap|''p'' : ''C'' → ''X''}}. Let ''c''<sub>1</sub> in ''C'' be in the fiber of ''x''. Then for every other ''c''<sub>2</sub> in the fiber of ''x'', there is precisely one deck transformation that takes ''c''<sub>1</sub> to ''c''<sub>2</sub>. This deck transformation corresponds to a curve ''g'' in ''C'' connecting ''c''<sub>1</sub> to ''c''<sub>2</sub>.
==Relations with groupoids==
One of the ways of expressing the algebraic content of the theory of covering spaces is using [[groupoids]] and the [[fundamental groupoid]]. The latter functor gives an equivalence of categories
between the category of covering spaces of a reasonably nice space ''X'' and the category of groupoid covering morphisms of π<sub>1</sub>(''X''). Thus a particular kind of ''map'' of spaces is well modelled by a particular kind of ''morphism'' of groupoids. The category of covering morphisms of a groupoid ''G'' is also equivalent to the category of actions of ''G'' on sets, and this allows the recovery of more traditional classifications of coverings. Proofs of these facts are given in the book 'Topology and Groupoids' referenced below.
==Relations with classifying spaces and group cohomology==
If ''X'' is a connected [[cell complex]] with [[homotopy group]]s {{nowrap|1=π<sub>''n''</sub>(''X'') = 0}} for all {{nowrap|''n'' ≥ 2}}, then the universal covering space ''T'' of ''X'' is contractible, as follows from applying the [[Whitehead theorem]] to ''T''. In this case ''X'' is a [[classifying space]] or {{nowrap|''K''(''G'', 1)}} for {{nowrap|1=''G'' = π<sub>''1''</sub>(''X'')}}.
Moreover, for every {{nowrap|''n'' ≥ 0}} the group of cellular ''n''-chains ''C''<sub>''n''</sub>(''T'') (that is, a [[free abelian group]] with basis given by ''n''-cells in ''T'') also has a natural '''Z'''''G''-[[Module (mathematics)|module]] structure. Here for an ''n''-cell ''σ'' in ''T'' and for ''g'' in ''G'' the cell ''g'' σ is exactly the translate of σ by a covering transformation of ''T'' corresponding to ''g''. Moreover, ''C''<sub>''n''</sub>(''T'') is a [[Free module|free]] '''Z'''''G''-module with free '''Z'''''G''-basis given by representatives of ''G''-orbits of ''n''-cells in ''T''. In this case the standard topological chain complex
where ε is the [[augmentation map]], is a [[free resolution|free '''Z'''''G''-resolution]] of '''Z''' (where '''Z''' is equipped with the trivial '''Z'''''G''-module structure, {{nowrap|1=''gm'' = ''m''}} for every {{nowrap|''g'' ∈ ''G''}} and every {{nowrap|''m'' ∈ '''Z'''}}). This resolution can be used to compute [[group cohomology]] of ''G'' with arbitrary coefficients.
The method of Graham Ellis for computing group resolutions and other aspects of homological algebra, as shown in his paper in J. Symbolic Comp. and his web page listed below, is to build a universal cover of a prospective {{nowrap|''K''(''G'', 1)}} inductively at the same time as a contracting homotopy of this universal cover. It is the latter which gives the computational method.
==Generalizations==
As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is [[locally path-connected]]. However, when the deck transformation group is a [[topological group]] whose topology is not [[discrete topology|discrete]], difficulties arise. Some progress has been made for more complex spaces, such as the [[Hawaiian earring]]; see the references there for further information.
A number of these difficulties are resolved with the notion of ''semicovering'' due to Jeremy Brazas, see the paper cited below. Every covering map is a semicovering, but semicoverings satisfy the "2 out of 3" rule: given a composition {{nowrap|1=''h'' = ''fg''}} of maps of spaces, if two of the maps are semicoverings, then so also is the third. This rule does not hold for coverings, since the composition of covering maps need not be a covering map.
Another generalisation is to actions of a group which are not free. Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of [[orbit space]]s wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." At present, "Topology and Groupoids" listed below seems to be the only basic topology text to cover such results.
==Applications==
[[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map {{nowrap|''T''<sup>3</sup> → '''RP'''<sup>3</sup>}} is not a covering map. In particular, the relevant map carries any element of ''T''<sup>3</sup>, that is, an ordered triple (a,b,c) of angles (real numbers mod 2π), to the composition of the three coordinate axis rotations R<sub>x</sub>(a)∘R<sub>y</sub>(b)∘R<sub>z</sub>(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group '''SO'''(3), which is topologically '''RP'''<sup>3</sup>.
This animation shows a set of three gimbals mounted together to allow ''three'' degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in ''gimbal lock''. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]
An important practical application of covering spaces occurs in [[charts on SO(3)]], the [[rotation group SO(3)|rotation group]]. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in [[navigation]], [[nautical engineering]], and [[aerospace engineering]], among many other uses. Topologically, SO(3) is the [[real projective space]] '''RP'''<sup>3</sup>, with fundamental group '''Z'''/2, and only (non-trivial) covering space the hypersphere ''S''<sup>3</sup>, which is the group [[spin group|Spin(3)]], and represented by the unit [[quaternions]]. Thus quaternions are a preferred method for representing spatial rotations – see [[quaternions and spatial rotation]].
However, it is often desirable to represent rotations by a set of three numbers, known as [[Euler angles]] (in numerous variants), both because this is conceptually simpler, and because one can build a combination of three [[gimbal]]s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''<sup>3</sup> of three angles to the real projective space '''RP'''<sup>3</sup> of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as [[gimbal lock]], and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the [[Rank (differential topology)|rank]] of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.
==See also==
*[[Bethe lattice]] is the universal cover of a [[Cayley graph]]
*[[Covering graph]], a covering space for an [[undirected graph]], and its special case the [[bipartite double cover]].
*[[Covering group]]
*[[Galois connection]]
==Notes==
==Notes==
<references/>
{{reflist}}
==References==
*{{cite book |ref=harv | first = Ronald | last = Brown| authorlink=Ronald Brown (mathematician) | title = Topology and Groupoids| year = 2006 | publisher = Booksurge LLC | location = Charleston, S. Carolina | isbn = 1-4196-2722-8 | url = http://www.bangor.ac.uk/r.brown/topgpds.html}} See chapter 10.
*{{SpringerEOM
| title=Covering
| id=Covering&oldid=13627
| last=Chernavskii
| first=A.V.
}}
*{{cite book | ref=harv | first = Hershel M. | last = Farkas | coauthors = [[Irwin Kra]] | title = Riemann Surfaces | year = 1980 | edition = 2nd ed. | publisher = Springer | location = New York | isbn = 0-387-90465-4}} See chapter 1 for a simple review.
*{{cite book | ref=harv | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
*[http://138.73.27.39/tac/reprints/articles/7/tr7abs.html Categories and groupoids, P.J. Higgins], downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
*{{cite book | ref=harv | first = Jurgen | last = Jost | title = Compact Riemann Surfaces | year = 2002 | publisher = Springer | location = New York | isbn = 3-540-43299-X}} See section 1.3
*{{cite book | ref=harv | first = William | last = Massey | authorlink = William S. Massey | title = A Basic Course in Algebraic Topology | year = 1991 | publisher = Springer | location = New York | isbn = 0-387-97430-X}} See chapter 5.
*{{cite book|ref=harv|last=Munkres|first=James R.|authorlink=James Munkres|title=Topology|year=2000|publisher=Prentice Hall|location=Upper Saddle River, NJ|isbn=0131816292|edition=2. ed.}}
* Brazas, J., 'Semicoverings: a generalisation of covering space theory', Homology, Homotopy, and Applications, 14 (2012), No. 1, pp. 33–63.
* Ellis, G. [http://hamilton.nuigalway.ie/Hap/www/ "Homological Algebra Program]
* Ellis, G. Computing group resolutions, J. Symbolic Comput. 38 (2004) 1077–1118.
*{{cite book |last=Spanier |first=Edwin |title=Algebraic Topology |date=December 1994 |publisher=Springer |isbn=0-387-94426-5}}
== References ==
*Burnside, William (1897) ''[http://www.gutenberg.org/ebooks/40395 Theory of Groups of Finite Order]'', [[Cambridge University Press]], at [[Project Gutenberg]] and [https://archive.org/details/theorygroupsfin00burngoog here] at [[Archive.org]]. (This is the first edition; the introduction to the second edition contains Burnside's famous ''volte face'' regarding the utility of [[representation theory]].)
* {{citation | last=Frobenius |first=Ferdinand Georg |authorlink=Ferdinand Georg Frobenius |title=Ueber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul |journal=Crelle |volume=CI |year=1887 |page=288}}.
* {{Citation | last1=Neumann | first1=Peter M. | author1-link=Peter M. Neumann | title=A lemma that is not Burnside's | mr=562002 | year=1979 | journal=The Mathematical Scientist | issn=0312-3685 | volume=4 | issue=2 | pages=133–141}}.
* {{citation | last=Rotman |first=Joseph |title=An introduction to the theory of groups |publisher=Springer-Verlag |year=1995 |isbn=0-387-94285-8}}.
[[Category:Algebraic topology]]
[[Category:Lemmas]]
[[Category:Homotopy theory]]
[[Category:Group theory]]
[[Category:Fiber bundles]]
[[Category:Topological graph theory]]
Revision as of 02:52, 11 August 2014
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to Template:Harvtxt.[1]
In the following, let G be a finitegroup that acts on a setX. For each g in G let Xg denote the set of elements in X that are fixed byg. Burnside's lemma asserts the following formula for the number of orbits, denoted |X/G|:[2]
Thus the number of orbits (a natural number or +∞) is equal to the average number of points fixed by an element of G (which is also a natural number or infinity). If G is infinite, the division by |G| may not be well-defined; in this case the following statement in cardinal arithmetic holds:
The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows.
Let X be the set of 36 possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group G of the cube act on X in the natural manner. Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the fixed sets for the 24 elements of G.
Cube with coloured faces
one identity element which leaves all 36 elements of X unchanged
six 90-degree face rotations, each of which leaves 33 of the elements of X unchanged
three 180-degree face rotations, each of which leaves 34 of the elements of X unchanged
eight 120-degree vertex rotations, each of which leaves 32 of the elements of X unchanged
six 180-degree edge rotations, each of which leaves 33 of the elements of X unchanged
A detailed examination of these automorphisms may be found
here.
The average fix size is thus
Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in n colors is given by
Proof
The first step in the proof of the lemma is to re-express the sum over the group elements g ∈ G as an equivalent sum over the set elements x ∈ X:
(Here Xg = {x ∈ X | g.x = x} is the subset of all points of X fixed by g ∈ G, whereas Gx = {g ∈ G | g.x = x} is the stabilizer subgroup of G that fixes the point x ∈ X.)
The orbit-stabilizer theorem says that there is a natural bijection for each x ∈ X between the orbit of x, G.x = {g.x | g ∈ G} ⊆ X, and the set of left cosets G/Gx of its stabilizer subgroup Gx. With Lagrange's theorem this implies
Our sum over the set X may therefore be rewritten as
Finally, notice that X is the disjoint union of all its orbits in X/G, which means the sum over X may be broken up into separate sums over each individual orbit.
Putting everything together gives the desired result:
History: the lemma that is not Burnside's
William Burnside stated and proved this lemma, attributing it to Template:Harvnb in his 1897 book on finite groups. But, even prior to Frobenius, the formula was known to Cauchy in 1845. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy. Consequently, this lemma is sometimes referred to as the lemma that is not Burnside's.[3] (see also Stigler's law of eponymy). This is less ambiguous than it may seem: Burnside contributed many lemmas to this field.
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
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