{{Short description|Spectral sequence}} In [[mathematics]], the '''Adams spectral sequence''' is a [[spectral sequence]] introduced by {{harvs|txt|first=J. Frank | authorlink=Frank Adams|last=Adams|year=1958}} which computes the [[stable homotopy theory|stable homotopy groups]] of [[topological spaces]]. Like all spectral sequences, it is a computational tool; it relates [[homology (mathematics)|homology]] theory to what is now called [[stable homotopy theory]]. It is a reformulation using [[homological algebra]], and an extension, of a technique called 'killing homotopy groups' applied by the French school of [[Henri Cartan]] and [[Jean-Pierre Serre]].
==Motivation==
For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be [[CW complex]]es. The [[singular cohomology|ordinary]] [[cohomology group]]s <math>H^*(X)</math> are understood to mean <math>H^*(X; \Z/p\Z)</math>.
The primary goal of [[algebraic topology]] is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is <math>S^n</math>, these maps form the ''n''th [[homotopy group]] of ''Y''. A more reasonable (but still very difficult!) goal is to understand the set <math>[X, Y]</math> of maps (up to homotopy) that remain after we apply the [[Suspension (topology)|suspension functor]] a large number of times. We call this the collection of stable maps from ''X'' to ''Y''. (This is the starting point of [[stable homotopy theory]]; more modern treatments of this topic begin with the concept of a [[Spectrum (homotopy theory)|spectrum]]. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.)
The set <math>[X, Y]</math> turns out to be an abelian group, and if ''X'' and ''Y'' are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime ''p''. In an attempt to compute the ''p''-torsion of <math>[X, Y]</math>, we look at cohomology: send <math>[X, Y]</math> to Hom(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')). This is a good idea because cohomology groups are usually tractable to compute.
The key idea is that <math>H^*(X)</math> is more than just a graded [[abelian group]], and more still than a graded [[ring (mathematics)|ring]] (via the [[cup product]]). The representability of the cohomology functor makes ''H''<sup>*</sup>(''X'') a [[module (mathematics)|module]] over the algebra of its stable [[cohomology operation]]s, the [[Steenrod algebra]] ''A''. Thinking about ''H''<sup>*</sup>(''X'') as an ''A''-module forgets some cup product structure, but the gain is enormous: Hom(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')) can now be taken to be ''A''-linear! A priori, the ''A''-module sees no more of [''X'', ''Y''] than it did when we considered it to be a map of vector spaces over F<sub>''p''</sub>. But we can now consider the derived functors of Hom in the category of ''A''-modules, [[Ext functor|Ext]]<sub>''A''</sub><sup>''r''</sup>(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')). These acquire a second grading from the grading on ''H''<sup>*</sup>(''Y''), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.
The point of all this is that ''A'' is so large that the above sheet of cohomological data contains all the information we need to recover the ''p''-primary part of [''X'', ''Y''], which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.
== Classical formulation ==
=== Formulation for computing homotopy groups of spectra === The classical Adams spectral sequence can be stated for any [[connective spectrum]] <math>X</math> of [[Spectrum of finite type|finite type]], meaning <math>\pi_i(X)=0</math> for <math>i < 0</math> and <math>\pi_i(X)</math> is a finitely generated Abelian group in each degree. Then, there is a spectral sequence <math>E_*^{*,*}(X)</math><ref name="Ravenel">{{Cite book|last=Ravenel|first=Douglas C.|url=https://web.math.rochester.edu/people/faculty/doug/mu.html|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1|location=Orlando|pages=|oclc=316566772|author-link=Douglas Ravenel}}</ref>{{rp|41}} such that
# <math>E_2^{s,t} = \text{Ext}_{A_p}^{s,t}(H^*(X), \Z/p)</math> for <math>A_p</math> the mod <math>p</math> Steenrod algebra # For <math>X</math> of finite type, <math>E_\infty^{*,*}</math> is a bigraded group associated with a filtration of <math>\pi_*(X)\otimes \Z_p</math> (the [[p-adic integers]])
Note that this implies for <math>X = \mathbb{S}</math>, this computes the <math>p</math>-torsion of the homotopy groups of the [[sphere spectrum]], i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex <math>Y</math> we can consider the suspension spectrum <math>\Sigma^\infty Y</math>, this gives the statement of the previous formulation as well.
This statement generalizes a little bit further by replacing the <math>\mathcal{A}_p</math>-module <math>\mathbb{Z}/p</math> with the cohomology groups <math>H^*(Y)</math> for some connective spectrum <math>Y</math> (or topological space <math>Y</math>). This is because the construction of the spectral sequence uses a "free" resolution of <math>H^*(X)</math> as an <math>\mathcal{A}_p</math>-module, hence we can compute the Ext groups with <math>H^*(Y)</math> as the second entry. We therefore get a spectral sequence with <math>E_2</math>-page given by<blockquote><math>E_2^{t,s} = \text{Ext}_{\mathcal{A}_p}^{s,t}(H^*(X),H^*(Y))</math></blockquote>which has the convergence property of being isomorphic to the graded pieces of a filtration of the <math>p</math>-torsion of the stable homotopy group of homotopy classes of maps between <math>X</math> and <math>Y</math>, that is<blockquote><math>E_2^{s,t} \Rightarrow \pi_{t-s}^{\mathbb{S}}([X,Y])\otimes \mathbb{Z}_{p}</math></blockquote>
=== Spectral sequence for the stable homotopy groups of spheres === For example, if we let both spectra be the sphere spectrum, so <math>X = Y = \mathbb{S}</math>, then the Adams spectral sequence has the convergence property<blockquote><math>E_2^{t,s} = \text{Ext}_{\mathcal{A}_p}^{s,t}(H^*(\mathbb{S}),H^*(\mathbb{S})) \Rightarrow \pi_{t-s}(\mathbb{S})\otimes \mathbb{Z}_p</math></blockquote>giving a technical tool for approaching a computation of the stable homotopy groups of spheres. It turns out that many of the first terms can be computed explicitly from purely algebraic information<ref name=":0">{{Cite web|last=Hatcher|first=Allen|date=|title=Spectral Sequences in Algebraic Topology|url=https://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf|url-status=live|archive-url=https://web.archive.org/web/20180728105526/http://pi.math.cornell.edu:80/~hatcher/SSAT/SSch2.pdf |archive-date=2018-07-28 |access-date=|website=}}</ref><sup>pp 23–25</sup>. Also note that we can rewrite <math>H^*(\mathbb{S}) = \mathbb{Z}/p</math>, so the <math>E_2</math>-page is<blockquote><math>E_2^{t,s} = \operatorname{Ext}_{\mathcal{A}_p}^{s,t}(\mathbb{Z}/p,\mathbb{Z}/p) \Rightarrow \pi_{t-s}(\mathbb{S})\otimes \mathbb{Z}_p</math></blockquote>We include this calculation information below for <math>p=2</math>.
=== Ext terms from the resolution === Given the [[Adams resolution]]<blockquote><math>\cdots \to H^*(F_2) \to H^*(F_1) \to H^*(F_0) \to H^*(X)</math></blockquote>we have the <math>E_1</math>-terms as<blockquote><math>E_1^{s,t} = \operatorname{Hom}^t_{\mathcal{A}_p}(H^*(F_s),H^*(Y))</math></blockquote>for the graded Hom-groups. Then the <math>E_1</math>-page can be written as<blockquote><math>E_1 = \begin{array}{c|ccc} 3 & \vdots & \vdots & \vdots\\ 2 & \text{Hom}^2(H^*(F_0), H^*(Y)) & \text{Hom}^2(H^*(F_1), H^*(Y)) & \text{Hom}^2(H^*(F_2), H^*(Y)) & \cdots \\ 1 & \text{Hom}^1(H^*(F_0), H^*(Y)) & \text{Hom}^1(H^*(F_1), H^*(Y)) & \text{Hom}^1(H^*(F_2), H^*(Y)) & \cdots \\ 0 & \text{Hom}^0(H^*(F_0), H^*(Y)) & \text{Hom}^0(H^*(F_1), H^*(Y)) & \text{Hom}^0(H^*(F_2), H^*(Y)) & \cdots \\ \hline & 0 & 1 & 2 \end{array}</math></blockquote>so the degree of <math>s</math> can be thought of how "deep" in the Adams resolution we go before we can find the generators.
== Calculations == The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.
=== Grading of the Differential === The <math>r</math>th Adams differential always goes to the left 1, and up <math>r</math>. That is, <blockquote><math>d_r \colon E_r^{s, t} \to E_r^{s-1,t+r}</math>.</blockquote> === Examples with Eilenberg–Maclane spectra === Some of the simplest calculations are with [[Eilenberg–Maclane spectrum|Eilenberg–Maclane spectra]] such as <math>X = H\Z</math> and <math>X = H\Z/(p^k)</math>.<ref name="Ravenel"/>{{rp|48}} For the first case, we have the <math>E_1</math> page<blockquote><math>E_1^{s,t} = \begin{cases} \Z/p & \text{ if } t = s \\ 0 & \text{ otherwise } \end{cases}</math></blockquote>giving a collapsed spectral sequence, hence <math>E_1 = E_\infty</math>. This can be rewritten as<blockquote><math>\text{Ext}^{s,t}_{\mathcal{A}_p}(H^*(H\Z), \Z/p) = \begin{cases} \Z/p & \text{ if } t = s \\ 0 & \text{ if } t \neq s \end{cases}</math></blockquote>giving the <math>E_2</math>-page. For the other case, note there is a cofiber sequence<blockquote><math>H\Z\xrightarrow{\cdot p^k} H\Z \to H\Z/p^k \to \Sigma H\Z</math></blockquote>which ends up giving a splitting in cohomology, so <math>H^*(H\Z/p^k) = H^*(H\Z)\oplus H^*(\Sigma H\Z)</math> as <math>\mathcal{A}_p</math>-modules. Then, the <math>E_2</math>-page of <math>H^*(H\Z/p)</math> can be read as<blockquote><math>E_2^{s,t} = \begin{cases} \Z/p & \text{if } t-s = 0,1 \\ 0 & \text{otherwise } \end{cases}</math></blockquote>The expected <math>E_\infty</math>-page is<blockquote><math>E_\infty^{s,t} = \begin{cases} \mathbb{Z}/p^k & \text{ if } t = s \\ 0 & \text{ otherwise } \end{cases}</math>.</blockquote>The only way for this spectral sequence to converge to this page is if is there are non-trivial differentials supported on every element with Adams grading <math>(s, s+1)</math>. === Other applications === Adams' original use for his spectral sequence was the first proof of the [[Hopf invariant]] 1 problem: <math>\R^n</math> admits a [[division algebra]] structure only for ''n'' = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in [[K-theory]].
The [[Thom space|Thom isomorphism theorem]] relates [[differential topology]] to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, [[John Milnor]] and [[Sergei Novikov (mathematician)|Sergei Novikov]] used the Adams spectral sequence to compute the coefficient ring of [[complex cobordism]]. Further, Milnor and [[C. T. C. Wall]] used the spectral sequence to prove Thom's conjecture on the structure of the oriented [[cobordism]] ring: two oriented manifolds are cobordant [[if and only if]] their [[Pontryagin class#Pontryagin numbers|Pontryagin]] and [[Stiefel–Whitney class#Stiefel–Whitney numbers|Stiefel–Whitney numbers]] agree.
== Stable homotopy groups of spheres == [[File:E 2 page of Adams spectral sequence of spheres for p = 2.png|thumb|609x609px|Visual diagram demonstrating the <math>E_2</math> page of the Adams spectral sequence computing the stable homotopy groups of spheres. The dots represent elements left over from the <math>E_1</math> page and the diagonal lines moving up and to the left represent various differentials in the spectral sequence. The differential <math>d_r</math> moves one unit to the left and <math>r</math> units upward. The vertical lines are used as a book-keeping tool for determining the structure of the torsion groups. Moreover, they represent multiplication by <math>2</math>. The lines moving up and right by one unit represent multiplication by <math>h_1</math>.]] Using the spectral sequence above for <math>X = Y = \mathbb{S}</math> we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres.<ref name=":0" /> For <math>p=2</math> this amounts to looking at the <math>E_2</math>-page with<blockquote><math>E_2^{s,t} = \text{Ext}_{\mathcal{A}_2}^{s,t}(\mathbb{Z}/2,\mathbb{Z}/2)</math></blockquote>This can be done by first looking at the Adams resolution of <math>\mathbb{Z}/2</math>. Since <math>\mathbb{Z}/2</math> is in degree <math>0</math>, we have a surjection<blockquote><math>\mathcal{A}_2\cdot \iota \to \mathbb{Z}/2</math></blockquote>where <math>\mathcal{A}_2</math> has a generator in degree <math>0</math> denoted <math>\iota</math>. The kernel <math>K_0</math> consists of all elements <math>Sq^I\iota</math> for admissible monomials <math>Sq^I</math> generating <math>\mathcal{A}_2</math>, hence we have a map<blockquote><math>\bigoplus_{I \text{ admissible}} \mathcal{A}_2\cdot Sq^I\iota \to K_0 </math></blockquote>and we denote each of the generators mapping to <math>Sq^i\iota</math> in the direct sum as <math>\alpha_i</math>, and the rest of the generators as <math>Sq^I\alpha_j</math> for some <math>j</math>. For example,<blockquote><math>\begin{align} \alpha_1 \mapsto Sq^1\iota & &Sq^2\alpha_1 \mapsto Sq^{2,1}\iota \\ \alpha_2 \mapsto Sq^2\iota & & Sq^1\alpha_2 \mapsto Sq^3\iota \\ \alpha_4 \mapsto Sq^4\iota & & Sq^3\alpha_1 \mapsto Sq^{3,1}\iota \\ \alpha_8 \mapsto Sq^8 & & Sq^2\alpha_2 \mapsto Sq^{3,1}\iota \end{align}</math></blockquote>Notice that the last two elements of <math>\alpha_i</math> map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as <math>Sq^1\alpha_1</math> since<blockquote><math>Sq^1\alpha_1 \mapsto Sq^1Sq^1\iota = 0</math></blockquote>because of the Adem relation. We call the generator of this element in <math>F_2</math>, <math>\beta_2</math>. We can apply the same process and get a kernel <math>K_1</math>, resolve it, and so on. When we do, we get an <math>E_1</math>-page which looks like<blockquote><math>E_1^{s,t} = \begin{array}{c|ccc} \vdots & \vdots & \vdots & \vdots\\ 4 & Sq^4\iota, Sq^{3,1}\iota & Sq^{2,1}\alpha_1, Sq^{3}\alpha_1, Sq^2\alpha_2, \alpha_4 & Sq^2\beta_2 & \cdots \\ 3 & Sq^3\iota, Sq^{2,1}\iota &Sq^2\alpha_1, Sq^1\alpha_2 & Sq^1\beta_2& \cdots \\ 2 & Sq^2\iota & \alpha_2, Sq^1\alpha_1 & \beta_2 & \cdots \\ 1 & Sq^1\iota & \alpha_1 & 0 & \cdots \\ 0 & \iota & 0 & 0 & \cdots \\ \hline & 0 & 1 & 2 \end{array}</math></blockquote> which can be expanded by computer up to degree <math>100</math> with relative ease. Using the found generators and relations, we can calculate the <math>E_2</math>-page with relative ease. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote <math>s</math> and the vertical index denote <math>t - s</math> giving a different type of diagram for the <math>E_2</math>-page<ref name=":0" /><sup>pg 21</sup>. See the diagram above for more information.
== Generalizations ==
The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by {{harvtxt|Novikov|1967}} where ordinary cohomology is replaced by a [[generalized cohomology theory]], often [[complex bordism]] or [[Brown–Peterson cohomology]]. This requires knowledge of the algebra of stable cohomology operations for the cohomology theory in question, but enables calculations which are completely intractable with the classical Adams spectral sequence.
== See also ==
* [[Postnikov system]] * [[Steenrod algebra]] * [[Spectrum (topology)]] *[[Adams resolution]] *[[Ravenel's conjectures]]
==References== {{refbegin}} *{{Citation | last1=Adams | first1=J. Frank | author-link=Frank Adams| title=On the structure and applications of the Steenrod algebra | doi=10.1007/BF02564578 | mr=0096219 | year=1958 | journal=[[Commentarii Mathematici Helvetici]] | issn=0010-2571 | volume=32 | issue=1 | pages=180–214| s2cid=15677036 }} *{{citation|mr=0185597|last=Adams|first= J. Frank| author-link=Frank Adams| title=Stable homotopy theory|publisher=Springer |series=Lecture Notes in Mathematics|volume=3 |orig-year=1964 |url=https://books.google.com/books?id=-vHtCAAAQBAJ |isbn=9783662159422 |year=2013}} *{{citation|last=Botvinnik|first=Boris|title=Manifolds with Singularities and the Adams–Novikov Spectral Sequence |series=London Mathematical Society Lecture Note Series|year=1992|isbn= 0-521-42608-1 |publisher=[[Cambridge University Press]] |url=https://books.google.com/books?id=a3AFRbZ1JnIC}} * {{Citation | last1=McCleary | first1=John | title=A User's Guide to Spectral Sequences | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-56759-6 | mr=1793722 |date=February 2001 | volume=58 | edition = 2nd}} *{{citation|first=Sergei|last=Novikov|author-link=Sergei Novikov (mathematician)|title=Methods of algebraic topology from the point of view of cobordism theory|journal= Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya|volume=31|year=1967|language=Russian|pages=855–951}} *{{Citation | last1=Ravenel | first1=Douglas C. |author-link=Douglas Ravenel| editor1-last=Barratt | editor1-first=M. G. | editor2-last=Mahowald | editor2-first=Mark E. | title=Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II | publisher=Springer | series=Lecture Notes in Mathematics | isbn=978-3-540-08859-2 | doi= 10.1007/BFb0068728 | mr=513586 | year=1978 | volume=658 | chapter=A novice's guide to the Adams–Novikov spectral sequence | pages=404–475}} * {{citation |last= Ravenel |first= Douglas C. |author-link=Douglas Ravenel |title= Complex cobordism and stable homotopy groups of spheres |edition= 2nd |url= http://www.math.rochester.edu/people/faculty/doug/mu.html |publisher= AMS Chelsea |year= 2003 |isbn= 978-0-8218-2967-7 |mr= 0860042 }}. {{refend}}
=== Overviews of computations === {{refbegin}} * {{cite arXiv |eprint=2001.04511 |title=More stable stems |first1=D.C. |last1=Isaksen |first2=G. |last2=Wang |first3=Z. |last3=Xu |date=2020|class=math.AT }} – computes all Adams spectral sequences for the stable homotopy groups of spheres up to degree 90 {{refend}}
=== Higher-order terms === {{refbegin}} *{{cite arXiv |eprint=math/0407045 |title=Computation of the E_3-term of the Adams spectral sequence |first1=H.J. |last1=Baues |first2=M. |last2=Jibladze|date=2004 }} *{{cite journal |arxiv=1108.3376 |title=Higher order derived functors and the Adams spectral sequence |first1=H.J. |last1=Baues |first2=D. |last2=Blanc |journal=Journal of Pure and Applied Algebra |volume=219 |issue=2 |date=2015 |pages=199–239 |doi=10.1016/j.jpaa.2014.04.018|s2cid=119144480 }} *{{cite journal |last1=Baues |first1=H.J. |last2=Frankland |first2=M. |title=2-track algebras and the Adams spectral sequence |journal=J. Homotopy Relat. Struct. |volume=11 |issue= 4|pages=679–713 |date=2016 |doi=10.1007/s40062-016-0147-x |arxiv=1505.03885|s2cid=119658430 }} {{refend}}
==External links== *{{citation|url=http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bruner-primer-2009.pdf |title=An Adams Spectral Sequence Primer|last=Bruner|first=Robert R.|date=June 2, 2009}} *{{citation|chapter-url=https://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf|title=Spectral Sequences|chapter= The Adams Spectral Sequence| last=Hatcher|first=Allen|author-link=Allen Hatcher}}
==Notes== {{Reflist}}
[[Category:Homotopy theory]] [[Category:Spectral sequences]]