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href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> <h4 id="stable_homotopy_theory">Stable Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#AdamsChart'>Adams Chart</a></li> <li><a href='#relation_to_brownpeterson_spectrum'>Relation to Brown-Peterson spectrum</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Adams-Novikov spectral sequence</em> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">E = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>.</p> <p>More in detail, the <em>Adams-Novikov spectral sequence</em> is a class of <a class="existingWikiWord" href="/nlab/show/spectral+sequences">spectral sequences</a>, which refine the <em><a class="existingWikiWord" href="/nlab/show/classical+Adams+spectral+sequence">classical Adams spectral sequence</a></em> by replacing <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/Fp">Fp</a> by <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>. This makes the Adams-Novikov spectral sequence converge to the full <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a>, hence in particular to the full <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> (<a href="#Novikov67">Novikov 67, Theorems 3.1, 3.3</a>, reviewed as <a href="#Ravenel78">Ravenel 78, Thm. 3.1</a>).</p> <p>More generally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> any suitable <a class="existingWikiWord" href="/nlab/show/E-infinity+ring">E-infinity ring</a> there is an Adams-Novikov-type spectral sequence involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a>/<a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a>. This fully general notion is often again just referred to as the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a></em>. Accordingly, see there for more. For detailed introduction, see at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+the+Adams+Spectral+Sequence">Introduction to the Adams Spectral Sequence</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="AdamsChart">Adams Chart</h3> <p><a href="#Isaksen14">Isaksen 14</a>:</p> <p><img src="/nlab/files/IsaksenANSSE2p.jpg" /></p> <p><img src="/nlab/files/IsaksenANSSEinfinityp.jpg" /></p> <h3 id="relation_to_brownpeterson_spectrum">Relation to Brown-Peterson spectrum</h3> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-component of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E^2</annotation></semantics></math>-term of the Adams-Novikov spectral sequence for the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, hence the one converging to the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\ast(\mathbb{S})</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ext</mi> <mrow><msub><mi>BP</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>BP</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>BP</mi> <mo>*</mo></msub><mo>,</mo><msub><mi>BP</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BP</mi></mrow><annotation encoding="application/x-tex">BP</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/Brown-Peterson+spectrum">Brown-Peterson spectrum</a> at prime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>recalled e.g. as <a href="#Ravenel">Ravenel, theorem 1.4.2</a></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chromatic+spectral+sequence">chromatic spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/EHP+spectral+sequence">EHP spectral sequence</a></p> </li> </ul> <h2 id="references">References</h2> <p>The generalization of the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">E = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/HA">HA</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">E = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/MU">MU</a> is due to</p> <ul> <li id="Novikov67"><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, <em>The methods of algebraic topology from the viewpoint of cobordism theories</em>, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian) (<a class="existingWikiWord" href="/nlab/files/Novikov67.pdf" title="Novikov67">Novikov67</a>)</li> </ul> <p>While that article was still being translated to English, <a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a> learned of and then lectured about this work in 1967 in Chicago. This lecture together with two later lectures in 1970 and in 1971 constitute the book</p> <ul> <li id="Adams74"><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalized+homology">Stable homotopy and generalized homology</a></em>, Chicago Lectures in mathematics, 1974</li> </ul> <p>Further early development of this Adams-Novikov spectral sequence:</p> <ul> <li id="Zahler72"><a class="existingWikiWord" href="/nlab/show/Raphael+Zahler">Raphael Zahler</a>, <em>The Adams-Novikov Spectral Sequence for the Spheres</em>, Annals of Mathematics Second Series, Vol. 96, No. 3 (Nov., 1972), pp. 480-504 (<a href="https://doi.org/10.2307/1970821">doi:10.2307/1970821</a>)</li> </ul> <p>Review:</p> <ul> <li id="Ravenel78"> <p><a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>A Novice’s guide to the Adams-Novikov spectral sequence</em>, in: <a class="existingWikiWord" href="/nlab/show/Michael+Barratt">Michael Barratt</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a> (eds.) <em>Geometric Applications of Homotopy Theory II</em>. Lecture Notes in Mathematics, vol 658, Springer 1978 (<a href="https://doi.org/10.1007/BFb0068728">doi:10.1007/BFb0068728</a>, <a href="https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf">pdf</a>)</p> </li> <li id="Ravenel"> <p><a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em>, page 11 of chapter I <em>An introduction to the homotopy groups of spheres</em> (<a href="http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel1.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, <em><a href="http://www.math.rochester.edu/people/faculty/doug/mu.html">Complex cobordism and stable homotopy groups of spheres</a></em>, chapter IV <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">B P</annotation></semantics></math>-Theory and the Adams-Novikov Spectral Sequence</em> (<a href="http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel4.pdf">pdf</a>)</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, sections 4.7 and 5.6 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Localizations and the Adams-Novikov Spectral Sequence</em> (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf">pdf</a>)</p> </li> </ul> <p>The Adams chart of the ANSS for large order has been presented in</p> <ul> <li id="Isaksen14"><a class="existingWikiWord" href="/nlab/show/Daniel+Isaksen">Daniel Isaksen</a>, <em>Classical and motivic Adams-Novikov charts</em> (<a href="http://arxiv.org/abs/1408.0248">arXiv:1408.0248</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 8, 2021 at 13:53:20. 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