|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the field of [[algebraic topology]], the '''Eilenberg–Moore spectral sequence''' addresses the calculation of the [[homology group]]s of a [[pullback]] over a [[fibration]]. The [[spectral sequence]] formulates the calculation from knowledge of the homology of the remaining spaces. [[Samuel Eilenberg]] and [[John C. Moore]]'s original paper addresses this for [[singular homology]].
| | Hi there, I am Alyson Boon although it is not the name on my beginning certification. Her family members life in Ohio. What me and my family members love is performing ballet but I've been taking on new things recently. My day occupation is an invoicing officer but I've currently applied for another 1.<br><br>Look into my web site; real psychics ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr]) |
| | |
| ==Motivation==
| |
| Let <math>k</math> be a [[field (mathematics)|field]] and
| |
| | |
| :<math>H_\ast(-)=H_\ast(-,k), H^\ast(-)=H^\ast(-,k)</math>
| |
| denote [[singular homology]] and [[singular cohomology]] with coefficients in ''k'', respectively.
| |
| | |
| Consider the following pullback ''E<sub>f</sub>'' of a continuous map ''p'':
| |
| :<math> \begin{array}{c c c} E_f &\rightarrow & E \\ \downarrow & & \downarrow{p}\\ X &\rightarrow_{ f} &B\\ \end{array} </math>
| |
| | |
| A frequent question is how the homology of the fiber product ''E<sub>f</sub>'', relates to the ones of ''B'', ''X'' and ''E''. For example, if ''B'' is a point, then the pullback is just the usual product ''E'' × ''X''. In this case the [[Künneth formula]] says
| |
| | |
| :''H''<sup>∗</sup>(''E<sub>f</sub>'') = ''H''<sup>∗</sup>(''X''×''E'') ≅ ''H''<sup>∗</sup>(''X'') ⊗<sub>k</sub> ''H''<sup>∗</sup>(''E'').
| |
| | |
| However this relation is not true in more general situations. The Eilenberg−Moore spectral sequence is a device which allows the computation of the (co)homology of the fiber product in certain situations.
| |
| | |
| ==Statement==
| |
| The Eilenberg−Moore spectral sequences generalizes the above isomorphism to the situation where ''p'' is a [[fibration]] of topological spaces and the base ''B'' is [[simply connected]]. Then there is a convergent spectral sequence with
| |
| :<math>E_2^{\ast,\ast}=\text{Tor}_{H^\ast(B)}^{\ast,\ast}(H^\ast(X),H^\ast(E))\Rightarrow H^\ast(E_f).</math>
| |
| This is a generalization insofar as the zeroeth [[Tor functor]] is just the tensor product and in the above special case the cohomology of the point ''B'' is just the coefficient field ''k'' (in degree 0).
| |
| | |
| Dually, we have the following homology spectral sequence:
| |
| :<math>E^2_{\ast,\ast}=\text{Cotor}^{H_\ast(B)}_{\ast,\ast}(H_\ast(X),H_\ast(E))\Rightarrow H_\ast(E_f).</math>
| |
| | |
| ==Indications on the proof==
| |
| The spectral sequence arises from the study of [[differential graded category|differential graded]] objects ([[chain complex]]es), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.
| |
| | |
| Let
| |
| :<math>S_\ast(-)=S_\ast(-,k)</math>
| |
| be the [[singular chain]] functor with coefficients in <math>k</math>. By the [[Eilenberg–Zilber theorem]], <math>S_\ast(B)</math> has a differential graded [[coalgebra]] structure over <math>k</math> with
| |
| structure maps
| |
| :<math>S_\ast(B)\xrightarrow{\triangle} S_\ast(B\times B)\xrightarrow{\simeq}S_\ast(B)\otimes S_\ast(B).</math>
| |
| | |
| In down-to-earth terms, the map assigns to a singular chain ''s'': ''Δ<sup>n</sup>'' → ''B'' the composition of ''s'' and the diagonal inclusion ''B'' ⊂ ''B'' × ''B''. Similarly, the maps <math>f</math> and <math>p</math> induce maps of differential graded coalgebras
| |
| | |
| <math>f_\ast \colon S_\ast(X)\rightarrow S_\ast(B)</math>, <math>p_\ast \colon S_\ast(E)\rightarrow S_\ast(B)</math>.
| |
| | |
| In the language of [[comodule]]s, they endow <math>S_\ast(E)</math> and <math>S_\ast(X)</math> with differential graded comodule structures over <math>S_\ast(B)</math>, with structure maps
| |
| | |
| :<math>S_\ast(X)\xrightarrow{\triangle} S_\ast(X)\otimes S_\ast(X)\xrightarrow{f_\ast\otimes 1} S_\ast(B)\otimes S_\ast(X)</math>
| |
| and similarly for ''E'' instead of ''X''. It is now possible to construct the so-called [[cobar resolution]] for
| |
| | |
| :<math>S_\ast(X)</math>
| |
| as a differential graded <math>S_\ast(B)</math> comodule. The cobar resolution is a standard technique in differential homological algebra:
| |
| | |
| :<math> \mathcal{C}(S_\ast(X),S_\ast(B))=\cdots\xleftarrow{\delta_2} \mathcal{C}_{-2}(S_\ast(X),S_\ast(B))\xleftarrow{\delta_1} \mathcal{C}_{-1}(S_\ast(X),S \ast(B))\xleftarrow{\delta_0} S_\ast(X)\otimes S_\ast(B),</math>
| |
| | |
| where the ''n''-th term <math>\mathcal{C}_{-n}</math> is given by
| |
| :<math>\mathcal{C}_{-n}(S_\ast(X),S_\ast(B))=S_\ast(X)\otimes \underbrace{S_\ast(B)\otimes \cdots \otimes S_\ast(B)}_{n}\otimes S_\ast(B).</math>
| |
| | |
| The maps <math>\delta_n</math> are given by
| |
| :<math>\lambda_f\otimes\cdots\otimes 1 + \sum_{i=2}^n 1\otimes\cdots \otimes\triangle_i\otimes\cdots\otimes 1,</math>
| |
| where <math>\lambda_f</math> is the structure map for <math>S_\ast(X)</math> as a left <math>S_\ast(B)</math> comodule.
| |
| | |
| The cobar resolution is a [[bicomplex]], one degree coming from the grading of the chain complexes ''S''<sub>∗</sub>(−), the other one is the simplicial degree ''n''. The [[total complex]] of the bicomplex is denoted <math>\mathbf{\mathcal{C}}_\bullet</math>.
| |
| | |
| The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
| |
| | |
| :<math>\Theta\colon \mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E)\rightarrow S_\ast(E_f,k)</math>
| |
| | |
| that induces a [[quasi-isomorphism]] (i.e. inducing an isomorphism on homology groups)
| |
| | |
| <center><math>\Theta_\ast \colon \operatorname{Cotor}^{S_\ast(B)}(S_\ast(X)S_\ast(E))\rightarrow H_\ast(E_f),</math></center>
| |
| | |
| where <math>\Box_{S_\ast(B)}</math> is the [[cotensor product]] and Cotor (cotorsion) is the
| |
| [[derived functor]] for the [[cotensor]] product.
| |
| | |
| To calculate
| |
| | |
| :<math>H_\ast(\mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E))</math>, | |
| | |
| view
| |
| | |
| :<math>\mathbf{\mathcal{C}}_{\bullet {\text{ }\Box_{S_\ast(B)}}}S_\ast(E)</math>
| |
| | |
| as a [[double complex]].
| |
| | |
| For any bicomplex there are two [[filtration]]s (see {{Harv|McCleary|2001}} or the [[spectral sequence]] of a filtered complex); in this case the Eilenberg−Moore spectral sequence results from filtering by increasing homological degree (by columns in the standard picture of a spectral sequence). This filtration yields
| |
| | |
| :<math>E^2=\operatorname{Cotor}^{H_\ast(B)}(H_\ast(X),H_\ast(E)).</math>
| |
| | |
| These results have been refined in various ways. For example {{Harv|Dwyer|1975}} refined the convergence results to include spaces for which
| |
| | |
| :<math>\pi_1(B)</math>
| |
| | |
| acts [[nilpotent]]ly on
| |
| | |
| :<math>H_i(E_f)</math>
| |
| | |
| for all <math>i\geq 0</math>
| |
| and {{Harv|Shipley|1996}} further generalized this to include arbitrary pullbacks.
| |
| | |
| The original construction does not lend itself to computations with other homology theories since there is no reason to expect that such a process would work for a homology theory not derived from chain complexes. However, it is possible to axiomatize the above procedure and give conditions under which the above spectral sequence holds for a general (co)homology theory, see Smith's original work {{Harv|Smith|1970}} or the introduction in {{Harv|Hatcher|2002}}).
| |
| | |
| ==References==
| |
| | |
| * {{Citation | last1=Dwyer | first1=William G. | title=Exotic convergence of the Eilenberg–Moore spectral sequence | year=1975 | journal=Illinois Journal of Mathematics | issn= 001-92082 | volume=19 | issue=4 | pages=607–617}}
| |
| * {{Citation | last1=Eilenberg | first1=Samuel | author1-link = Samuel Eilenberg | last2=Moore | first2=John C. | title=Limits and spectral sequences | year=1962 | journal=Topology. An International Journal of Mathematics | volume=1 | issue=1 | pages=1–23 | doi=10.1016/0040-9383(62)90093-9 }}
| |
| * {{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}
| |
| * {{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 | year=2001 | volume=58 |chapter=Chapters 7 and 8: The Eilenberg−Moore spectral sequence I and II}}
| |
| * {{Citation | last1=Shipley | first1=Brooke E. | title=Convergence of the homology spectral sequence of a cosimplicial space | year=1996 | journal=[[American Journal of Mathematics]] | volume=118 | issue=1 | pages=179–207 | doi=10.1353/ajm.1996.0004}}
| |
| * {{Citation | last1=Smith | first1=Larry | title=Lectures on the Eilenberg−Moore spectral sequence | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | mr=0275435 | year=1970 | volume=134}}
| |
| | |
| {{DEFAULTSORT:Eilenberg-Moore spectral sequence}}
| |
| [[Category:Spectral sequences]]
| |
| [[Category:Homology theory]]
| |
Hi there, I am Alyson Boon although it is not the name on my beginning certification. Her family members life in Ohio. What me and my family members love is performing ballet but I've been taking on new things recently. My day occupation is an invoicing officer but I've currently applied for another 1.
Look into my web site; real psychics (www.chk.woobi.co.kr)