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| In [[homotopy theory]], a branch of [[mathematics]], the '''Barratt–Priddy theorem''' (also referred to as '''Barratt–Priddy–Quillen theorem''') expresses a connection between the homology of the [[symmetric group]]s and mapping spaces of spheres. It is also often stated as a relation between the [[sphere spectrum]] and the [[classifying space]]s of the symmetric groups via Quillen's [[plus construction]].
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| ==Statement of the theorem==
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| The mapping space {{math|Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>)}} is the topological space of all continuous maps {{math|''f'': ''S''<sup>''n''</sup> → ''S''<sup>''n''</sup>}} from the {{mvar|n}}-dimensional sphere {{mvar|S<sup>n</sup>}} to itself, under the topology of [[uniform convergence]] (a special case of the [[compact-open topology]]). These maps are required to fix a basepoint {{math|''x'' ∈ ''S''<sup>''n''</sup>}}, satisfying {{math|''f''(''x''){{=}}''x''}}, and to have [[degree of a continuous mapping|degree]] 0; this guarantees that the mapping space is [[connected space|connected]]. The Barratt–Priddy theorem expresses a relation between the homology of these mapping spaces and the homology of the [[symmetric group]]s {{mvar|Σ<sub>n</sub>}}. | | The stylish wardrobe of Maggie Gyllenhaal�s role in BBC Two�s hard-hitting political thriller, The Honourable Woman, has caught the attention of the eagle-eyed viewers.<br><br>The eight-part series, set against the backdrop of the Israeli-Palestinian conflict, centres around Nessa Stein, played by Gyllenhaal. Stein is an Anglo-Israeli businesswoman recently ennobled in the House of Lords who devotes herself to philanthropic purposes across the Middle East, but hides a secret past from her time spent in Gaza eight years earlier.<br><br>Through the unravelling of her public and private life played out on an international, political stage, Stein parades in an increasingly impressive selection of outfits.<br>�Because the character of Nessa is so complicated and multi layered, we looked at all sorts of different people as reference. I suppose we started off by looking at other powerful and stylish women through history, Jackie Kennedy, Eva Peron, Margaret Thatcher, Cleopatra� Edward K Gibbon costume designer for the series told The Independent<br><br> |
| | Maggie Gyllenhaal The Honourable Woma<br> |
| | �And then we kind of threw all the reference away and started afresh. The way Maggie looked as Nessa was constantly evolving throughout the six month shoot.� |
| | The series opens with Nessa clad in a Roland Mouret power dress. Her day to day look is a sartorial dream with tailored suits by the likes of Stella McCartney, Acne, Escada, Pringle and vintage Chane<br> |
| | �Silk blouses and wide legged pants based on 1970s Yves Saint Laurent originals were created by Hilary Marschner� explains Gibb<br><br> |
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| It follows from the [[Freudenthal suspension theorem]] and the [[Hurewicz theorem]] that the {{mvar|k}}th [[homology theory|homology]] {{math|''H''<sub>''k''</sub>(Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>))}} of this mapping space is ''independent'' of the dimension {{mvar|n}}, as long as {{math|''n''>''k''}}. Similarly, Nakaoka (1960) proved that the {{mvar|k}}th [[group homology]] {{math|''H''<sub>''k''</sub>(''Σ''<sub>''n''</sub>)}} of the symmetric group {{mvar|Σ<sub>n</sub>}} on {{mvar|n}} elements is independent of {{mvar|n}}, as long as {{math|''n''≥2''k''}}.
| | Outerwear includes coats by Mulberry, vintage finds from Jil Sander and a 1980s Gieves and Hawkes men�s co<br>. |
| | Even curled up in her panic room at night she [http://Www.Squidoo.com/search/results?q=sports+silk sports silk] slips by haute couture Parisian lingerie designer Carine Gilson and London based lingerie label Bod<br>. |
| | In pictures: Nessa Stein's wardrobe in The Honourable Woma<br> |
| | Shoes are by Acne, Christian Louboutin and [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bags Outlet]. With bags from Mulberry and John Lewis. �Nessa's wardrobe runs the full gamut from designer, through High Street, Charity shops and bespoke pieces� says Gib<br><br> |
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| The Barratt–Priddy theorem states that these "stable homology groups" are the same: for {{math|''n''≥2''k''}} there is a natural isomorphism
| | �The clothing is always the way in [to the character]� Gyllenhaal told WWD. �I never played a character that didn�t care about what they were wearing.� |
| | | The Honourable Woman continues tonight, BBC2 at 9pm. |
| :<math>H_k(\Sigma_n)\cong H_k(\text{Map}_0(S^n,S^n))</math>
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| This isomorphism holds with integral coefficients (in fact with any coefficients, as is made clear in the reformulation below).
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| ==Example: first homology==
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| This isomorphism can be seen explicitly for the first homology {{math|''H''<sub>1</sub>}}. The [[Group cohomology#H1|first homology of a group]] is the largest [[abelian group|commutative]] quotient of that group. For the permutation groups {{mvar|Σ<sub>n</sub>}}, the only commutative quotient is given by the [[sign of a permutation]], taking values in {{math|{−1, 1}}}. This shows that {{math|''H''<sub>1</sub>(''Σ''<sub>''n''</sub>)≅'''Z'''/2'''Z'''}}, the [[cyclic group]] of order 2, for all {{math|''n''≥2}}. (For {{math|''n''{{=}}1}}, {{math|''Σ''<sub>1</sub>}} is the trivial group, so {{math|''H''<sub>1</sub>(''Σ''<sub>''n''</sub>){{=}}0}}.)
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| It follows from the theory of [[covering space]]s that the mapping space {{math|Map<sub>0</sub>(''S''<sup>1</sup>,''S''<sup>1</sup>)}} of the circle {{math|''S''<sup>1</sup>}} is [[contractible space|contractible]], so
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| {{math|''H''<sub>1</sub>(Map<sub>0</sub>(''S''<sup>1</sup>,''S''<sup>1</sup>)){{=}}0}}. For the 2-sphere {{math|''S''<sup>2</sup>}}, the first [[homotopy group]] and first homology group of the mapping space are [[Homotopy_groups_of_spheres#.CF.803.28S2.29_.3D_Z|both infinite cyclic]]:
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| {{math|π<sub>1</sub>(Map<sub>0</sub>(''S''<sup>2</sup>,''S''<sup>2</sup>))≅''H''<sub>1</sub>(Map<sub>0</sub>(''S''<sup>2</sup>,''S''<sup>2</sup>))≅'''Z'''}}. A generator for this group can be built from the [[Hopf fibration]] {{math|''S''<sup>3</sup>→''S''<sup>2</sup>}}. Finally, once {{math|''n''≥3}}, both are [[Homotopy_groups_of_spheres#Table_of_homotopy_groups|cyclic of order 2]]:
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| {{math|π<sub>1</sub>(Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>))≅''H''<sub>1</sub>(Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>))}}≅'''Z'''/2'''Z'''.
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| ==Reformulation of the theorem==
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| The infinite symmetric group {{math|''Σ''<sub>∞</sub>}} is the union of the finite [[symmetric group]]s {{math|''Σ''<sub>''n''</sub>}}, and Nakaoka's theorem implies that the group homology of {{math|''Σ''<sub>∞</sub>}} is the stable homology of {{math|''Σ''<sub>''n''</sub>}}: for {{math|''n''≥2''k''}}, {{math|''H''<sub>''k''</sub>(''Σ''<sub>∞</sub>)≅''H''<sub>''k''</sub>(''Σ''<sub>''n''</sub>)}}.
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| The [[classifying space]] of this group is denoted {{math|''BΣ''<sub>∞</sub>}}, and its homology of this space is the group homology of {{math|''Σ''<sub>∞</sub>}}: {{math|''H''<sub>''k''</sub>(''BΣ''<sub>∞</sub>)≅''H''<sub>''k''</sub>(''Σ''<sub>∞</sub>)}}.
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| We similarly denote by {{math|Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}} the union of the mapping spaces {{math|Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>)}} (under the inclusions induced by [[suspension (topology)|suspension]]). The homology of {{math|Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}} is the stable homology of the previous mapping spaces: for {{math|''n''>''k''}}, {{math|''H''<sub>''k''</sub>(Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>))≅''H''<sub>''k''</sub>(Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>))}}.
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| There is a natural map {{math|''φ'': ''BΣ''<sub>∞</sub>→Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}} (one way to construct {{mvar|φ}} is via the model of {{math|''BΣ''<sub>∞</sub>}} as the space of finite subsets of {{math|'''R'''<sup>∞</sup>}} endowed with a certain topology). An equivalent formulation of the Barratt–Priddy theorem is that {{mvar|φ}} is a ''homology equivalence'' (or ''acyclic map''), meaning that {{mvar|φ}} induces an isomorphism on all homology groups with any local coefficient system.
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| ==Relation with Quillen's plus construction==
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| The Barratt–Priddy theorem implies that the space {{math|''BΣ''<sub>∞</sub><sup>+</sup>}} resulting from applying Quillen's [[plus construction]] to {{math|''BΣ''<sub>∞</sub>}} can be identified with {{math|Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}}. (Since {{math|π<sub>1</sub>(Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>))≅''H''<sub>1</sub>(''Σ''<sub>∞</sub>)≅'''Z'''/2'''Z'''}}, the map {{math|''φ'': ''BΣ''<sub>∞</sub>→Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}} satisfies the universal property of the plus construction once it is known that {{mvar|φ}} is a homology equivalence.)
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| The mapping spaces {{math|Map<sub>0</sub>(''S''<sup>''n''</sup>,''S''<sup>''n''</sup>)}} are more commonly denoted by {{math|''Ω''<sup>''n''</sup><sub>0</sub>''S''<sup>''n''</sup>}}, where {{math|''Ω''<sup>''n''</sup>''S''<sup>''n''</sup>}} is the {{mvar|n}}-fold [[loop space]] of the {{mvar|n}}-sphere {{mvar|S<sup>n</sup>}}, and similarly {{math|Map<sub>0</sub>(''S''<sup>∞</sup>,''S''<sup>∞</sup>)}} is denoted by {{math|''Ω''<sup>∞</sup><sub>0</sub>''S''<sup>∞</sup>}}. Therefore the Barratt–Priddy theorem can also be stated as
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| :<math>B\Sigma_\infty^+\simeq \Omega_0^\infty S^\infty</math> or
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| :<math>\textbf{Z}\times B\Sigma_\infty^+\simeq \Omega^\infty S^\infty</math>
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| In particular, the homotopy groups of {{math|''BΣ''<sub>∞</sub><sup>+</sup>}} are the [[homotopy groups of spheres|stable homotopy groups of spheres]]:
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| :<math>\pi_i(B\Sigma_\infty^+)\cong \pi_i(\Omega^\infty S^\infty)\cong \lim_{n\rightarrow \infty} \pi_{n+i}(S^n)=\pi_i^s</math>
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| =="''K''-theory of '''F'''<sub>1</sub>"==
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| The Barratt–Priddy theorem is sometimes colloquially rephrased as saying that "the ''K''-groups of '''F'''<sub>1</sub> are the stable homotopy groups of spheres". This is not a meaningful mathematical statement, but a metaphor expressing an analogy with [[algebraic K-theory|algebraic ''K''-theory]].
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| The "[[field with one element]]" '''F'''<sub>1</sub> is not a mathematical object; it refers to a collection of analogies between algebra and combinatorics. One central analogy is the idea that {{math|''GL''<sub>''n''</sub>('''F'''<sub>1</sub>)}} should be the symmetric group {{math|''Σ''<sub>''n''</sub>}}. | |
| The [[Algebraic_K-theory#Higher_K-theory|higher ''K''-groups]] {{math|''K''<sub>''i''</sub>(''R'')}} of a ring ''R'' can be defined as
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| :<math>K_i(R)=\pi_i(BGL_\infty(R)^+)</math>
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| According to this analogy, the K-groups {{math|''K''<sub>''i''</sub>('''F'''<sub>1</sub>)}} of {{math|'''F'''<sub>1</sub>}} should be defined as {{math|π<sub>''i''</sub>(''BGL''<sub>∞</sub>('''F'''<sub>1</sub>)<sup>+</sup>){{=}}π<sub>''i''</sub>(''BΣ''<sub>∞</sub><sup>+</sup>)}}, which by the Barratt–Priddy theorem is:
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| :<math>K_i(\mathbf{F}_1)=\pi_i(BGL_\infty(\mathbf{F}_1)^+)=\pi_i(B\Sigma_\infty^+)=\pi_i^s.</math>
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| ==References==
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| * {{Citation | last1=Barratt | first1=Michael | last2=Priddy | first2=Stewart | title=On the homology of non-connected monoids and their associated groups | url=http://www.digizeitschriften.de/dms/img/?PPN=PPN358147735_0047&DMDID=dmdlog5 | year=1972 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=47 | pages=1–14}}
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| * {{Citation
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| | last = Nakaoka
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| | first = Minoru
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| | year = 1960
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| | title = Decomposition theorem for homology groups of symmetric groups
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| | journal = Annals of Mathematics
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| | volume = 71
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| | pages = 16–42
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| | url = http://www.jstor.org/stable/1969878
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| | mr = 112134
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| }}
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| {{DEFAULTSORT:Barratt-Priddy theorem}}
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| [[Category:Homotopy theory]]
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The stylish wardrobe of Maggie Gyllenhaal�s role in BBC Two�s hard-hitting political thriller, The Honourable Woman, has caught the attention of the eagle-eyed viewers.
The eight-part series, set against the backdrop of the Israeli-Palestinian conflict, centres around Nessa Stein, played by Gyllenhaal. Stein is an Anglo-Israeli businesswoman recently ennobled in the House of Lords who devotes herself to philanthropic purposes across the Middle East, but hides a secret past from her time spent in Gaza eight years earlier.
Through the unravelling of her public and private life played out on an international, political stage, Stein parades in an increasingly impressive selection of outfits.
�Because the character of Nessa is so complicated and multi layered, we looked at all sorts of different people as reference. I suppose we started off by looking at other powerful and stylish women through history, Jackie Kennedy, Eva Peron, Margaret Thatcher, Cleopatra� Edward K Gibbon costume designer for the series told The Independent
Maggie Gyllenhaal The Honourable Woma
�And then we kind of threw all the reference away and started afresh. The way Maggie looked as Nessa was constantly evolving throughout the six month shoot.�
The series opens with Nessa clad in a Roland Mouret power dress. Her day to day look is a sartorial dream with tailored suits by the likes of Stella McCartney, Acne, Escada, Pringle and vintage Chane
�Silk blouses and wide legged pants based on 1970s Yves Saint Laurent originals were created by Hilary Marschner� explains Gibb
Outerwear includes coats by Mulberry, vintage finds from Jil Sander and a 1980s Gieves and Hawkes men�s co
.
Even curled up in her panic room at night she sports silk slips by haute couture Parisian lingerie designer Carine Gilson and London based lingerie label Bod
.
In pictures: Nessa Stein's wardrobe in The Honourable Woma
Shoes are by Acne, Christian Louboutin and Celine Bags Outlet. With bags from Mulberry and John Lewis. �Nessa's wardrobe runs the full gamut from designer, through High Street, Charity shops and bespoke pieces� says Gib
�The clothing is always the way in [to the character]� Gyllenhaal told WWD. �I never played a character that didn�t care about what they were wearing.�
The Honourable Woman continues tonight, BBC2 at 9pm.