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At least for <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> this is known as the <em><a class="existingWikiWord" href="/nlab/show/Pontrjagin+ring">Pontrjagin ring</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_*(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToWhiteheadProduct">Relation to Whitehead product</h3> <p>Under the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>, the <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a> of the Pontrjagin product on homology is the <a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of a <a class="existingWikiWord" href="/nlab/show/based+loop+space">based loop space</a>.</p> <p>This is due to <a href="#Samelson53">Samelson (1953)</a> and for higher Whitehead brackets due to <a href="#Arkowitz71">Arkowitz (1971)</a>.</p> <p>In fact, in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+ring">Pontrjagin ring</a> structure on <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/based+loop+spaces">based loop spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> is identified via the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a> with the <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a> (see <a href="universal+enveloping+algebra#ForSuperLieAlgebras">there</a>) of the <a class="existingWikiWord" href="/nlab/show/Whitehead+bracket+super+Lie+algebra">Whitehead bracket super Lie algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> [<a href="#MilnorMoore65">Milnor & Moore (1965) Appendix</a>; <a href="#Whitehead78">Whitehead (1978) Thm. X.7.10</a>; <a href="#FélixHalperinThomas00">Félix, Halperin & Thomas 2000, Thm. 16.13</a>]. Moreover, in this case the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (<a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">hence</a> the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>) <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> may be read off from any <a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan model</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (by the proposition <a href="Sullivan+model+of+loop+space#SullivanModelForBasedLoopSpace">here</a>).</p> <p>For the following examples we use these notational conventions:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math> denotes a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>,</p> </li> <li> <p>a subscript on a generator denotes its degree,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">⟨</mo><mi>⋯</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}\langle\cdots \rangle</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <a class="existingWikiWord" href="/nlab/show/linear+span">spanned</a> by the <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> of generators listed inside the angular brackets;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">[</mo><mi>⋯</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}[\cdots]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> of the free <a class="existingWikiWord" href="/nlab/show/graded-commutative+algebra">graded-commutative algebra</a> on the set of generators listed inside the square brackets;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(-)</annotation></semantics></math> denotes the graded <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> on a given <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a>,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/semifree+dgc-algebra">semifree dgc-algebra</a> on a given set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>⋯</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\cdots\}</annotation></semantics></math> of generators subject to differential relations we denote, with slight abuse of notation, by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">[</mo><mi>⋯</mi><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center"><mtr><mtd><mi mathvariant="normal">d</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>…</mi></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbb{K}[\cdots]\big/ \left( \begin{array}{c} \mathrm{d}(-) = \ldots \\ \vdots \end{array} \right) </annotation></semantics></math></div></li> </ul> <p> <div class='num_remark' id='RationalPontrAlgOfLoopsOfSpheresAndCPn'> <h6>Example</h6> <p><strong>(rational Pontrjagin algebra of loops of spheres and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^n</annotation></semantics></math>s)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+spheres">Sullivan model of</a> the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mn>0</mn></mtd></mtr> <mtr><mtd><mi mathvariant="normal">d</mi><msub><mi>ω</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ω</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub></mtd></mtr></mtable><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{l} S^2) \;\simeq\; \mathbb{Q}\big[ \omega_2, \omega_3 \big]\big/ \left( \begin{array}{l} \mathrm{d} \omega_2 \,=\, 0 \\ \mathrm{d} \omega_3 \,=\, \tfrac{1}{2} \omega_2 \wedge \omega_2 \end{array} \right) \,. </annotation></semantics></math></div> <p>From this it follows (since <a class="existingWikiWord" href="/nlab/show/the+co-binary+Sullivan+differential+is+the+dual+Whitehead+product">the co-binary Sullivan differential is the dual Whitehead product</a>) that the binary <a class="existingWikiWord" href="/nlab/show/Whitehead+brackets">Whitehead</a> <a class="existingWikiWord" href="/nlab/show/super+Lie+brackets">super Lie brackets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> are:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>v</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{l} [v_1, v_1] = v_2 \\ [v_2, v_2] = 0 \\ [v_1, v_2] = 0 \end{array} </annotation></semantics></math></div> <p>The rational Pontrjagin ring of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Omega S^2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a> of this <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, hence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ω</mi><msup><mi>S</mi> <mn>2</mn></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕂</mi><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mn>2</mn><msubsup><mi>v</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>−</mo><msub><mi>v</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>v</mi> <mn>1</mn></msub><msub><mi>v</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>v</mi> <mn>2</mn></msub><msub><mi>v</mi> <mn>1</mn></msub></mtd></mtr></mtable><mo>)</mo></mrow><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> H_\bullet\big( \Omega S^2 ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_1, v_2\rangle \big) \big/ \left( \begin{array}{l} 2 v_1^2 - v_2 \\ v_1 v_2 - v_2 v_1 \end{array} \right) \mathrlap{\,.} </annotation></semantics></math></div> <p>Therefore the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> of the Pontrjagin ring of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Omega S^2</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}[v_1, v_2]</annotation></semantics></math> but the product of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_1</annotation></semantics></math> is deformed from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v_1 \cdot v_1 = 0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>v</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">v_1 \cdot v_1 = \tfrac{1}{2}v_2</annotation></semantics></math>.</p> <p>Similarly, the rational Pontrjagin algebra of the loop space of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ω</mi><msup><mi>S</mi> <mn>4</mn></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕂</mi><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>6</mn></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mn>2</mn><msubsup><mi>v</mi> <mn>3</mn> <mn>2</mn></msubsup><mo>−</mo><msub><mi>v</mi> <mn>6</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>v</mi> <mn>3</mn></msub><msub><mi>v</mi> <mn>6</mn></msub><mo>−</mo><msub><mi>v</mi> <mn>6</mn></msub><msub><mi>v</mi> <mn>3</mn></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> H_\bullet\big( \Omega S^4 ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_3, v_6\rangle \big) \big/ \left( \begin{array}{l} 2 v_3^2 - v_6 \\ v_3 v_6 - v_6 v_3 \end{array} \right) </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>6</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}[v_3, v_6]</annotation></semantics></math> but with the product of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">v_3</annotation></semantics></math> deformed from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>3</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v_3 \cdot v_3 = 0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>3</mn></msub><mo>⋅</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>v</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">v_3 \cdot v_3 = \tfrac{1}{2}v_6</annotation></semantics></math>.</p> <p>On the other hand, the differential of the <a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+complex+projective+space">Sullivan model of</a> <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> has vanishing co-binary part, so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ω</mi><mi>ℂ</mi><msup><mi>P</mi> <mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕂</mi><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mn>2</mn><msubsup><mi>v</mi> <mn>1</mn> <mn>2</mn></msubsup></mtd></mtr> <mtr><mtd><msub><mi>v</mi> <mn>1</mn></msub><msub><mi>v</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>−</mo><msub><mi>v</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><msub><mi>v</mi> <mn>1</mn></msub></mtd></mtr></mtable><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>𝕂</mi><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> H_\bullet\big( \Omega \mathbb{C}P^{n \geq 2} ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_2, v_{2n}\rangle \big) \big/ \left( \begin{array}{l} 2 v_1^2 \\ v_1 v_{2n} - v_{2n} v_1 \end{array} \right) \;\simeq\; \mathbb{K}[v_1, v_{2n}] </annotation></semantics></math></div> <p>is just a plain graded-commutative algebra.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n = \infty</annotation></semantics></math> (ie. for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>≃</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}P^\infty \simeq</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/BU%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B U(1)</annotation> </semantics> </math></a>) this becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ω</mi><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕂</mi><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>2</mn><msubsup><mi>v</mi> <mn>1</mn> <mn>2</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>𝕂</mi><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> H_\bullet\big( \Omega \mathbb{C}P^{\infty} ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_2\rangle \big) \big/ \big( 2 v_1^2 \big) \;\simeq\; \mathbb{K}[v_1] \mathrlap{\,,} </annotation></semantics></math></div> <p>reflecting the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>Ω</mi><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Omega \, \mathbb{C}P^\infty \,\simeq\, \Omega B U(1)\,\simeq\, S^1</annotation></semantics></math>.</p> </div> </p> <h3 id="homological_group_completion">Homological group completion</h3> <p>The homological version of the <em><a class="existingWikiWord" href="/nlab/show/group+completion+theorem">group completion theorem</a></em> relates the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+ring">Pontrjagin ring</a> of a <a class="existingWikiWord" href="/nlab/show/topological+monoid">topological monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to that of its <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>B</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Omega B A</annotation></semantics></math>.</p> <h3 id="RelationToQuantumCohomology">Relation to quantum cohomology</h3> <p>The homology Pontrjagin product of certain <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> is identified with the <a class="existingWikiWord" href="/nlab/show/quantum+cohomology">quantum cohomology</a> of corresponding <a class="existingWikiWord" href="/nlab/show/flag+varieties">flag varieties</a> (see references <a href="#ReferencesOnQuantumCohomologyAsPontrjaginRings">below</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homology+of+loop+spaces">homology of loop spaces</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The concept and the terminology “Pontryagin-multiplication” is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Hans+Samelson">Hans Samelson</a>, <em>On the Pontryagin product in spaces of paths</em>, Commentarii Mathematici Helvetici <strong>27</strong> (1953) 320–337 [<a href="https://doi.org/10.1007/BF02564566">doi:10.1007/BF02564566</a>]</li> </ul> <p>who name it in honor of the analogous product operation on the homology of <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em>Homologies in compact Lie groups</em>, Rec. Math. [Mat. Sbornik] N. S. <strong>6</strong> (<strong>48</strong>) 3 (1939) 389–422 [<a href="http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5835&option_lang=eng">mathnet:5835</a>]</li> </ul> <p>see also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans+Samelson">Hans Samelson</a>, §3 in: <em>Beiträge Zur Topologie der Gruppen-Mannigfaltigkeiten</em>, Annals of Mathematics, Second Series, <strong>42</strong> 5 (1941) 1091-1137 [<a href="https://www.jstor.org/stable/1970463">jsotr:1970463</a>]</li> </ul> <p>Proof that the <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a> of the Pontrjagin product is the <a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a>, under the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>:</p> <ul> <li id="Samelson53"><a class="existingWikiWord" href="/nlab/show/Hans+Samelson">Hans Samelson</a>, <em>A Connection Between the Whitehead and the Pontryagin Product</em>, American Journal of Mathematics <strong>75</strong> 4 (1953) 744–752 [<a href="https://doi.org/10.2307/2372549">doi:10.2307/2372549</a>]</li> </ul> <p>and in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>:</p> <ul> <li id="MilnorMoore65"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/John+Moore">John Moore</a>, Appendix (pp. 262) of: <em>On the structure of Hopf algebras</em>, Annals of Math. <strong>81</strong> (1965) 211-264 [<a href="https://doi.org/10.2307/1970615">doi:10.2307/1970615</a>, <a href="http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf">pdf</a>]</p> </li> <li id="FélixHalperinThomas00"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Claude+Thomas">Jean-Claude Thomas</a>, Thm. 16.13 in: <em>Rational Homotopy Theory</em>, Graduate Texts in Mathematics <strong>205</strong> Springer (2000) [<a href="https://link.springer.com/book/10.1007/978-1-4613-0105-9">doi:10.1007/978-1-4613-0105-9</a>]</p> </li> </ul> <p>and slightly beyond</p> <ul> <li id="Halperin92"> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <em>Universal enveloping algebras and loop space homology</em>, Journal of Pure and Applied Algebra <strong>83</strong> 3 (1992) 237-282 [<a href="https://doi.org/10.1016/0022-4049(92)90046-I">doi:10.1016/0022-4049(92)90046-I</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+A.+Scott">Jonathan A. Scott</a>, <em>Algebraic Structure in the Loop Space Homology Bockstein Spectral Sequence</em>, Transactions of the American Mathematical Society <strong>354</strong> 8 (2002) 3075-3084 [<a href="https://www.jstor.org/stable/3073034">jstor:3073034</a>]</p> </li> </ul> <p>and for higher Whitehead brackets:</p> <ul> <li id="Arkowitz71"><a class="existingWikiWord" href="/nlab/show/Martin+Arkowitz">Martin Arkowitz</a>, <em>Whitehead Products as Images of Pontrjagin Products</em>, Transactions of the American Mathematical Society, <strong>158</strong> 2 (1971) 453-463 [<a href="https://doi.org/10.2307/1995917">doi:10.2307/1995917</a>]</li> </ul> <p>and full lifting of the theorem <a href="#MilnorMoore65">Milnor & Moore, 1965 (Appendix)</a>, equipping the rational Pontrjagin algebra with <a class="existingWikiWord" href="/nlab/show/A-infinity+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>A</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">A_\infty</annotation> </semantics> </math>-algebra</a> structure and identifying it with the universal envelope of the <a class="existingWikiWord" href="/nlab/show/Whitehead+L-infinity+algebra">Whitehead L-infinity algebra</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Moreno-Fern%C3%A1ndez">José M. Moreno-Fernández</a>, Thm. 4.1 in: <em>The Milnor-Moore theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras in rational homotopy theory</em>, Mathematische Zeitschrift <strong>300</strong> (2022) 2147–2165 [<a href="https://arxiv.org/abs/1904.12530">arXiv:1904.12530</a>, <a href="https://doi.org/10.1007/s00209-021-02838-z">doi:10.1007/s00209-021-02838-z</a>]</li> </ul> <p>Refinement to algebra structure on the <a class="existingWikiWord" href="/nlab/show/singular+chain+complex">singular chain complex</a> (Adams-Hilton model):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+F.+Adams">John F. Adams</a>, <a class="existingWikiWord" href="/nlab/show/Peter+J.+Hilton">Peter J. Hilton</a>, <em>On the chain algebra of a loop space</em>, Commentarii Mathematici Helvetici <strong>30</strong> (1956) 305–330 [<a href="https://doi.org/10.1007/BF02564350">doi:10.1007/BF02564350</a>]</li> </ul> <p>reviewed and further developed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, <a class="existingWikiWord" href="/nlab/show/Paul-Eug%C3%A8ne+Parent">Paul-Eugène Parent</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Scott">Jonathan Scott</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Tonks">Andrew Tonks</a>, <em>A canonical enriched Adams-Hilton model for simplicial sets</em>, Advances in Mathematics <strong>207</strong> 2 (2006) 847-875 [<a href="https://doi.org/10.1016/j.aim.2006.01.013">doi:10.1016/j.aim.2006.01.013</a>, <a href="https://arxiv.org/abs/math/0408216">arXiv:math/0408216</a>]</li> </ul> <p>More on the Pontrjagin rings of the <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ichiro+Yokota">Ichiro Yokota</a>, <em>On the homology of classical Lie groups</em>, J. Inst. Polytech. Osaka City Univ. Ser. A <strong>8</strong> 2 (1957) 93-120 [<a href="https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-8/issue-2/On-the-homology-of-classical-Lie-groups/ojm/1353054814.full">euclid:ojm/1353054814</a>]</li> </ul> <p>reviewed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Inna+Zakharevich">Inna Zakharevich</a>, pp. 35 of: <em>Stable homotopy</em>, Section 11 of: <em><a href="https://pi.math.cornell.edu/~zakh/6530/">K-theory and characteristic classes</a></em>, lecture notes (2017) [<a href="https://pi.math.cornell.edu/~zakh/6530/sec11.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Zakharevich-StableHomotopy.pdf" title="pdf">pdf</a>]</li> </ul> <p>Further early discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <em>The space of loops on a Lie group</em>, Michigan Math. J. <strong>5</strong> 1 (1958) 35-61 [<a href="https://projecteuclid.org/journals/michigan-mathematical-journal/volume-5/issue-1/The-space-of-loops-on-a-Lie-group/10.1307/mmj/1028998010.full">doi:10.1307/mmj/1028998010</a>]</p> <blockquote> <p>(for loop spaces of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Browder">William Browder</a>, <em>Homology and Homotopy of H-Spaces</em>, Proceedings of the National Academy of Sciences of the United States of America <strong>46</strong> 4 (1960) 543-545 [<a href="https://www.jstor.org/stable/70867">jstor:70867</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Browder">William Browder</a>, p. 36 of: <em>Torsion in H-Spaces</em>, Annals of Mathematics, Second Series <strong>74</strong> 1 (1961) 24-51 [<a href="https://www.jstor.org/stable/1970305">jstor:1970305</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Browder">William Browder</a>, <em>Homology Rings of Groups</em>, American Journal of Mathematics <strong>90</strong> 1 (1968) [<a href="https://www.jstor.org/stable/2373440">jstor:2373440</a>]</p> </li> </ul> <p>On the effect on Pontrjagin rings of <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> of <a class="existingWikiWord" href="/nlab/show/topological+monoids">topological monoids</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barratt">Michael Barratt</a>, <a class="existingWikiWord" href="/nlab/show/Stewart+Priddy">Stewart Priddy</a>, <em>On the homology of non-connected monoids and their associated groups</em>, Commentarii Mathematici Helvetici, <strong>47</strong> 1 (1972) 1–14 [<a href="https://link.springer.com/article/10.1007/BF02566785">doi:10.1007/BF02566785</a>, <a href="https://eudml.org/doc/139496">eudml:139496</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>: <em>Homology fibrations and the “group-completion” theorem</em>, Inventiones mathematicae <strong>31</strong> (1976) 279-284 [<a href="https://doi.org/10.1007/BF01403148">doi:10.1007/BF01403148</a>]</p> </li> </ul> <p>Pontrjagin rings in the context of <a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Moira+Chas">Moira Chas</a>, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, §3 in: <em>String Topology</em> [<a href="https://arxiv.org/abs/math/9911159">arXiv:math/9911159</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+L.+Cohen">Ralph L. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+D.+S.+Jones">John D. S. Jones</a>, Jun Yan, pp. 15 of: <em>The loop homology algebra of spheres and projective spaces</em>, in <em>Categorical Decomposition Techniques in Algebraic Topology</em>, Progress in Mathematics <strong>215</strong>, Birkhäuser (2003) [<a href="https://arxiv.org/abs/math/0210353">arXiv:math/0210353</a>, <a href="https://doi.org/10.1007/978-3-0348-7863-0_5">doi:10.1007/978-3-0348-7863-0_5</a>]</p> </li> <li> <p>Gwénaël Massuyeau, <a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Brackets in the Pontryagin algebras of manifolds</em>, Mém. Soc. Math. France <strong>154</strong> (2017) [<a href="https://arxiv.org/abs/1308.5131">arXiv:1308.5131</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="Whitehead78"> <p><a class="existingWikiWord" href="/nlab/show/George+W.+Whitehead">George W. Whitehead</a>, p. 98 in: <em>Elements of Homotopy Theory</em>, Springer (1978) [<a href="https://link.springer.com/book/10.1007/978-1-4612-6318-0">doi:10.1007/978-1-4612-6318-0</a>]</p> </li> <li id="Hatcher02"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, §3.C, pp. 287 in: <em>Algebraic Topology</em>, Cambridge University Press (2002) [<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401">ISBN:9780521795401</a>, <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">webpage</a>]</p> </li> </ul> <p>Lecture notes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tyrone+Cutler">Tyrone Cutler</a>, §3 in: <em>The Bott-Samelson Theorem</em> (2020) [<a href="https://www.math.uni-bielefeld.de/~tcutler/pdf/Week%2011%20-%20The%20Bott-Samelson%20Theorem.pdf">pdf</a>]</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Pontryagin_product">Pontryagin product</a></em></li> </ul> <p>Relating the Pontrjagin algebra on <a class="existingWikiWord" href="/nlab/show/loop+groups">loop groups</a> of <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> to their <a class="existingWikiWord" href="/nlab/show/Langlands+dual+groups">Langlands dual groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zhiwei+Yun">Zhiwei Yun</a>, <a class="existingWikiWord" href="/nlab/show/Xinwen+Zhu">Xinwen Zhu</a>, <em>Integral homology of loop groups via Langlands dual groups</em>, Represent. Theory <strong>15</strong> (2011) 347-369 [<a href="https://arxiv.org/abs/0909.5487">arXiv:0909.5487</a>, <a href="https://doi.org/10.1090/S1088-4165-2011-00399-X">doi:10.1090/S1088-4165-2011-00399-X</a>]</li> </ul> <p>Computation of the Pontryagin products for (loop spaces of) <a class="existingWikiWord" href="/nlab/show/flag+manifolds">flag manifolds</a>:</p> <ul> <li>Jelena Grbic, Svjetlana Terzic, <em>The integral Pontrjagin homology of the based loop space on a flag manifold</em>, Osaka J. Math. <strong>47</strong> (2010) 439-460 [<a href="https://arxiv.org/abs/math/0702113">arXiv:math/0702113</a></li> </ul> <div> <h3 id="ReferencesOnQuantumCohomologyAsPontrjaginRings">Quantum cohomology as Pontrjagin rings</h3> <p>On the relation between <a class="existingWikiWord" href="/nlab/show/quantum+cohomology+rings">quantum cohomology rings</a>, hence of <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+invariants">Gromov-Witten invariants</a> in <a class="existingWikiWord" href="/nlab/show/topological+string+theory">topological string theory</a>, for <a class="existingWikiWord" href="/nlab/show/flag+manifold">flag manifold</a> <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> (such as the <a class="existingWikiWord" href="/nlab/show/CP%5E1+sigma-model"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℂ</mi> <msup><mi>P</mi> <mn>1</mn></msup> </mrow> <annotation encoding="application/x-tex">\mathbb{C}P^1</annotation> </semantics> </math>-sigma model</a>) and <a class="existingWikiWord" href="/nlab/show/Pontrjagin+rings">Pontrjagin rings</a> (<a class="existingWikiWord" href="/nlab/show/homology">homology</a>-<a class="existingWikiWord" href="/nlab/show/Hopf+algebras">Hopf algebras</a> of <a class="existingWikiWord" href="/nlab/show/based+loop+spaces">based loop spaces</a>):</p> <p>That the <a class="existingWikiWord" href="/nlab/show/Pontryagin+ring">Pontryagin ring</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> on the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of the <a class="existingWikiWord" href="/nlab/show/based+loop+space">based loop space</a> of a <a class="existingWikiWord" href="/nlab/show/simply-connected+topological+space">simply-connected</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is essentially the <a class="existingWikiWord" href="/nlab/show/quantum+cohomology+ring">quantum cohomology ring</a> of the <a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a> of its <a class="existingWikiWord" href="/nlab/show/complexification">complexification</a> by its <a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a> is attributed (“Peterson isomorphism”) to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dale+H.+Peterson">Dale H. Peterson</a> (notes by <a class="existingWikiWord" href="/nlab/show/Arun+Ram">Arun Ram</a>), <em>Quantum Cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></em>, MIT (1997) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html">web</a>, <a href="http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Peterson-QuantumCohomology.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>see also</p> <ul> <li>Yasha Savelyev, <em>Quantum characteristic classes and the Hofer metric</em>, Geom. Topol. <strong>12</strong> (2008) 2277-2326 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/0709.4510">arXiv:0709.4510</a>, <a href="https://doi.org/10.2140/gt.2008.12.2277">doi:10.2140/gt.2008.12.2277</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>and proven in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Lam">Thomas Lam</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Shimozono">Mark Shimozono</a>, §6.2 in: <em>Quantum cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">G/P</annotation></semantics></math> and homology of affine Grassmannian</em>, Acta Mathematica <strong>204</strong> (2010) 49–90 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/0705.1386">arXiv:0705.1386</a>, <a href="https://doi.org/10.1007/s11511-010-0045-8">doi:10.1007/s11511-010-0045-8</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chi+Hong+Chow">Chi Hong Chow</a>, <em>Peterson-Lam-Shimozono’s theorem is an affine analogue of quantum Chevalley formula</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2110.09985">arXiv:2110.09985</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Chow22"> <p><a class="existingWikiWord" href="/nlab/show/Chi+Hong+Chow">Chi Hong Chow</a>, <em>On D. Peterson’s presentation of quantum cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">G/P</annotation></semantics></math></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2210.17382">arXiv:2210.17382</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>reviewed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jimmy+Chow">Jimmy Chow</a>, <em>Homology of based loop groups and quantum cohomology of flag varieties</em>, talk at <em>Western Hemisphere Virtual Symplectic Seminar</em> (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://freemath.xyz/chowslides.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Chow-QuantumCohomologyOfFlagVarieties.pdf" title="pdf">pdf</a>, video:<a href="https://www.youtube.com/watch?v=ovbQBj7t3eU">YT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>with further discussion in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Naichung+Conan+Leung">Naichung Conan Leung</a>, <a class="existingWikiWord" href="/nlab/show/Changzheng+Li">Changzheng Li</a>, <em>Gromov-Witten invariants for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">G/B</annotation></semantics></math> and Pontryagin product for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">\Omega K</annotation></semantics></math></em>, Transactions of the American Mathematical Society <strong>364</strong> 5 (2012) 2567-2599 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.ams.org/journals/tran/2012-364-05/S0002-9947-2012-05438-9/">web</a>, <a href="https://arxiv.org/abs/0810.4859">arXiv:0810.4859</a>, <a href="https://www.jstor.org/stable/41524936">jstor:41524936</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>On the variant for Pontryagin products not on <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> but in <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological</a> <a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a>:</p> <ul> <li id="LamLiMihalceaShimozono18"> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Lam">Thomas Lam</a>, <a class="existingWikiWord" href="/nlab/show/Changzheng+Li">Changzheng Li</a>, <a class="existingWikiWord" href="/nlab/show/Leonardo+C.+Mihalcea">Leonardo C. Mihalcea</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Shimozono">Mark Shimozono</a>, <em>A conjectural Peterson isomorphism in K-theory</em>, Journal of Algebra <strong>513</strong> (2018) 326-343 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/j.jalgebra.2018.07.029">doi:10.1016/j.jalgebra.2018.07.029</a>, <a href="https://arxiv.org/abs/1705.03435">arXiv:1705.03435</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p>Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno, <em>Peterson Isomorphism in K-theory and Relativistic Toda Lattice</em>, International Mathematics Research Notices <strong>2020</strong> 19 (2020) 6421–6462 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1703.08664">arXiv:1703.08664</a>, <a href="https://doi.org/10.1093/imrn/rny051">doi:10.1093/imrn/rny051</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Syu+Kato">Syu Kato</a>, <em>Loop structure on equivariant K-theory of semi-infinite flag manifolds</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1805.01718">arXiv:1805.01718</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Kato19"> <p><a class="existingWikiWord" href="/nlab/show/Syu+Kato">Syu Kato</a>, <em>On quantum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-groups of partial flag manifolds</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1906.09343">arXiv:1906.09343</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Syu+Kato">Syu Kato</a>, <em>Darboux coordinates on the BFM spaces</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2008.01310">arXiv:2008.01310</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Kato21"> <p><a class="existingWikiWord" href="/nlab/show/Syu+Kato">Syu Kato</a>, <em>Quantum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-groups on flag manifolds</em>, talk at <em>IMPANGA 20</em> (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.impan.pl/konferencje/bcc/2020/20-impanga/impanga2021.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>On the example of the <a class="existingWikiWord" href="/nlab/show/CP%5E1+sigma-model">CP^1 sigma-model</a>: <a href="#LamLiMihalceaShimozono18">LLMS18, §4.1</a>, <a href="#Kato21">Kato21 p. 17</a>, <a href="#Chow22">Chow22 Exp. 1.4</a>.</p> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Gauge theory and mirror symmetry</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1404.6305">arXiv:1404.6305</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/chiral+rings">chiral rings</a> of <a class="existingWikiWord" href="/nlab/show/D%3D3+N%3D4+super+Yang-Mills+theory">D=3 N=4 super Yang-Mills theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Coulomb branches for quaternionic representations</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2209.01088">arXiv:2209.01088</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on January 28, 2025 at 07:42:24. 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