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nLab > Latest Changes: Spin(4)

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMar 6th 2019
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Author: Urs
Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/Spin%284%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Spin%284%29">current</a>

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v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMar 6th 2019
- (edited Mar 7th 2019)
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Author: Urs
Format: MarkdownItexgave this its own little entry in order to have a place where to record the following fact: *** The [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] of [[Spin(3)]] is [[polynomial ring|freely generated]] from $1/4$th of the [[first Pontryagin class]]: $$ H^\bullet \big( B Spin(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{4}p_1 \big] $$ Moreover, the [[integral cohomology|integral]] [[cohomology ring]] of the [[classifying space]] of [[Spin(4)]] is [[polynomial ring|freely generated]] from the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1$ and the combination $\tfrac{1}{2}\big( e + \tfrac{1}{2}p_1 \big)$, where $\chi$ is the [[Euler class]]: $$ H^\bullet \big( B Spin(4), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2}p_1 \,, \tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big) \big] $$ Finally, under the exceptional isomorphism (eq:Spin3SquareToO4) $\vartheta \;\colon\; Spin(3) \times Spin(3) \overset{\simeq}{\to} Spin(4)$ these classes are related by $$ \begin{aligned} \vartheta^\ast \left( \tfrac{1}{2}p_1 \right) & = \phantom{-} \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \\ \vartheta^\ast \Big( \tfrac{1}{2} \big( \chi + \tfrac{1}{2} p_1 \big) \Big) & = \phantom{-} \phantom{ 1 \otimes \tfrac{1}{4}p_1 + } 1 \otimes \tfrac{1}{4}p_1 \\ \text{hence} \phantom{AAAA} \vartheta^\ast\big( \chi \big) \phantom{ + \tfrac{1}{2} p_1 \big) \Big) } & = - \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \end{aligned} $$ Therefore, under the canonical [[diagonal]] inclusion $\iota \colon Spin(3) \overset{\Delta}{\hookrightarrow} Spin(3) \times Spin(3) \simeq Spin(4)$ (eq:Spin3Diagonally) we have $$ \begin{aligned} \iota^\ast \left( \tfrac{1}{2}p_1 \right) & = \tfrac{1}{2}p_1 \\ \iota^\ast \big( \chi \big) & = 0 \end{aligned} $$ <a href="https://ncatlab.org/nlab/revision/Spin%284%29/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Spin%284%29">current</a>

gave this its own little entry in order to have a place where to record the following fact:

The integral cohomology ring of the classifying space of Spin(3) is freely generated from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</annotation></semantics></math>$ th of the first Pontryagin class:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \big( B Spin(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{4}p_1 \big] </annotation></semantics></math>$
Moreover, the integral cohomology ring of the classifying space of Spin(4) is freely generated from the first fractional Pontryagin class $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}p_1</annotation></semantics></math>$ and the combination $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>e</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\big( e + \tfrac{1}{2}p_1 \big)</annotation></semantics></math>$ , where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math>$ is the Euler class:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mspace width="0.16667em"/><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \big( B Spin(4), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2}p_1 \,, \tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big) \big] </annotation></semantics></math>$
Finally, under the exceptional isomorphism (eq:Spin3SquareToO4) $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϑ</mi><mspace width="0.27778em"/><mo lspace="0.11111em">:</mo><mspace width="0.27778em"/><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vartheta \;\colon\; Spin(3) \times Spin(3) \overset{\simeq}{\to} Spin(4)</annotation></semantics></math>$ these classes are related by
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mi>ϑ</mi> <mo>*</mo></msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mphantom><mo lspace="0.11111em" rspace="0em">−</mo></mphantom><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo>⊗</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>⊗</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mi>ϑ</mi> <mo>*</mo></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd> <mtd><mo>=</mo><mphantom><mo lspace="0.11111em" rspace="0em">−</mo></mphantom><mphantom><mrow><mn>1</mn><mo>⊗</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo></mrow></mphantom><mn>1</mn><mo>⊗</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><mtext>hence</mtext><mphantom><mi>AAAA</mi></mphantom><msup><mi>ϑ</mi> <mo>*</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mphantom><mrow><mo lspace="0.11111em" rspace="0em">+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow></mphantom></mtd> <mtd><mo>=</mo><mo lspace="0.11111em" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo>⊗</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>⊗</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \vartheta^\ast \left( \tfrac{1}{2}p_1 \right) &amp; = \phantom{-} \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \\ \vartheta^\ast \Big( \tfrac{1}{2} \big( \chi + \tfrac{1}{2} p_1 \big) \Big) &amp; = \phantom{-} \phantom{ 1 \otimes \tfrac{1}{4}p_1 + } 1 \otimes \tfrac{1}{4}p_1 \\ \text{hence} \phantom{AAAA} \vartheta^\ast\big( \chi \big) \phantom{ + \tfrac{1}{2} p_1 \big) \Big) } &amp; = - \tfrac{1}{4}p_1 \otimes 1 + 1 \otimes \tfrac{1}{4} p_1 \end{aligned} </annotation></semantics></math>$
Therefore, under the canonical diagonal inclusion $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo lspace="0.11111em">:</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mover><mo>↪</mo><mi>Δ</mi></mover><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota \colon Spin(3) \overset{\Delta}{\hookrightarrow} Spin(3) \times Spin(3) \simeq Spin(4)</annotation></semantics></math>$ (eq:Spin3Diagonally) we have
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mi>ι</mi> <mo>*</mo></msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo>)</mo></mrow></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mi>ι</mi> <mo>*</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \iota^\ast \left( \tfrac{1}{2}p_1 \right) &amp; = \tfrac{1}{2}p_1 \\ \iota^\ast \big( \chi \big) &amp; = 0 \end{aligned} </annotation></semantics></math>$
v1, current
- CommentRowNumber3.
- CommentAuthorDavidRoberts
- CommentTimeMar 6th 2019
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Author: DavidRoberts
Format: MarkdownItexIs $e=\chi$?

Is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mo>=</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">e=\chi</annotation></semantics></math>$ ?
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeMar 7th 2019
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Author: Urs
Format: MarkdownItexThanks for catching. Fixed now.

Thanks for catching. Fixed now.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeMar 22nd 2019
- (edited Mar 22nd 2019)
- PermaLink
Author: Urs
Format: MarkdownItexQuestion: Might the second generator in $$ H^\bullet \big( B Spin(4), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2}p_1 \,, \tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big) \big] $$ actually come from $B SO(4)$? In other words, is the rational combination $\tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big)$ possibly already integral for $SO(4)$-bundles, not just for $Spin(4)$-bundles? [edit: ah, no, this cannot be...]

Question:

Might the second generator in
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mspace width="0.16667em"/><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \big( B Spin(4), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2}p_1 \,, \tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big) \big] </annotation></semantics></math>$
actually come from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B SO(4)</annotation></semantics></math>$ ?

In other words, is the rational combination $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>χ</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\big( \chi+ \tfrac{1}{2}p_1 \big)</annotation></semantics></math>$ possibly already integral for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(4)</annotation></semantics></math>$ -bundles, not just for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(4)</annotation></semantics></math>$ -bundles?

[edit: ah, no, this cannot be…]
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeAug 29th 2019
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Author: Urs
Format: MarkdownItexadded picture of D3 and A3 Dynkin diagrams with their central node removed, illustrating the exceptional iso $Spin(4) \simeq SU(2)\times SU(2)$. <a href="https://ncatlab.org/nlab/revision/diff/Spin%284%29/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%284%29/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Spin%284%29">current</a>

added picture of D3 and A3 Dynkin diagrams with their central node removed, illustrating the exceptional iso $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(4) \simeq SU(2)\times SU(2)</annotation></semantics></math>$ .

diff, v12, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJun 19th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Marcel Berger]], Section 8.9 of: *Geometry I*, Springer 1987 ([doi:10.1007/978-3-540-93815-6](https://doi.org/10.1007/978-3-540-93815-6)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%284%29/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%284%29/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Spin%284%29">current</a>
added pointer to:
- Marcel Berger, Section 8.9 of: Geometry I, Springer 1987 (doi:10.1007/978-3-540-93815-6)
diff, v13, current

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