Gaisser–Hillas function: Difference between revisions

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In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups 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.

Motivation

Let $k$ be a field and

H_{*} (-) = H_{*} (-, k), H^{*} (-) = H^{*} (-, k)

denote singular homology and singular cohomology with coefficients in k, respectively.

Consider the following pullback E_f of a continuous map p:

\begin{array}{ccc} E_{f} & \to & E \\ ↓ & ↓ p \\ X & \to_{f} & B \end{array}

A frequent question is how the homology of the fiber product E_f, 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^∗(E_f) = H^∗(X×E) ≅ H^∗(X) ⊗_k H^∗(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

E_{2}^{*, *} = {Tor}_{H^{*} (B)}^{*, *} (H^{*} (X), H^{*} (E)) \Rightarrow H^{*} (E_{f}) .

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:

E_{*, *}^{2} = {Cotor}_{*, *}^{H_{*} (B)} (H_{*} (X), H_{*} (E)) \Rightarrow H_{*} (E_{f}) .

Indications on the proof

The spectral sequence arises from the study of differential graded objects (chain complexes), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.

Let

S_{*} (-) = S_{*} (-, k)

be the singular chain functor with coefficients in $k$ . By the Eilenberg–Zilber theorem, $S_{*} (B)$ has a differential graded coalgebra structure over $k$ with structure maps

S_{*} (B) \overset{△}{\to} S_{*} (B \times B) \overset{≃}{\to} S_{*} (B) \otimes S_{*} (B) .

In down-to-earth terms, the map assigns to a singular chain s: Δⁿ → B the composition of s and the diagonal inclusion B ⊂ B × B. Similarly, the maps $f$ and $p$ induce maps of differential graded coalgebras

$f_{*} : S_{*} (X) \to S_{*} (B)$ , $p_{*} : S_{*} (E) \to S_{*} (B)$ .

In the language of comodules, they endow $S_{*} (E)$ and $S_{*} (X)$ with differential graded comodule structures over $S_{*} (B)$ , with structure maps

S_{*} (X) \overset{△}{\to} S_{*} (X) \otimes S_{*} (X) \overset{f_{*} \otimes 1}{\to} S_{*} (B) \otimes S_{*} (X)

and similarly for E instead of X. It is now possible to construct the so-called cobar resolution for

S_{*} (X)

as a differential graded $S_{*} (B)$ comodule. The cobar resolution is a standard technique in differential homological algebra:

𝒞 (S_{*} (X), S_{*} (B)) = \dots \overset{δ_{2}}{\leftarrow} 𝒞_{- 2} (S_{*} (X), S_{*} (B)) \overset{δ_{1}}{\leftarrow} 𝒞_{- 1} (S_{*} (X), S * (B)) \overset{δ_{0}}{\leftarrow} S_{*} (X) \otimes S_{*} (B),

where the n-th term $𝒞_{- n}$ is given by

𝒞_{- n} (S_{*} (X), S_{*} (B)) = S_{*} (X) \otimes \underset{n}{\underset{⏟}{S_{*} (B) \otimes \dots \otimes S_{*} (B)}} \otimes S_{*} (B) .

The maps $δ_{n}$ are given by

λ_{f} \otimes \dots \otimes 1 + \sum_{i = 2}^{n} 1 \otimes \dots \otimes △_{i} \otimes \dots \otimes 1,

where $λ_{f}$ is the structure map for $S_{*} (X)$ as a left $S_{*} (B)$ comodule.

The cobar resolution is a bicomplex, one degree coming from the grading of the chain complexes S_∗(−), the other one is the simplicial degree n. The total complex of the bicomplex is denoted $𝒞_{∙}$ .

The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map

Θ : 𝒞_{∙ ◻_{S_{*} (B)}} S_{*} (E) \to S_{*} (E_{f}, k)

that induces a quasi-isomorphism (i.e. inducing an isomorphism on homology groups)

Θ_{*} : {Cotor}^{S_{*} (B)} (S_{*} (X) S_{*} (E)) \to H_{*} (E_{f}),

where $◻_{S_{*} (B)}$ is the cotensor product and Cotor (cotorsion) is the derived functor for the cotensor product.

To calculate

H_{*} (𝒞_{∙ ◻_{S_{*} (B)}} S_{*} (E))

,

view

𝒞_{∙ ◻_{S_{*} (B)}} S_{*} (E)

as a double complex.

For any bicomplex there are two filtrations (see Template:Harv 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

E^{2} = {Cotor}^{H_{*} (B)} (H_{*} (X), H_{*} (E)) .

These results have been refined in various ways. For example Template:Harv refined the convergence results to include spaces for which

π_{1} (B)

acts nilpotently on

H_{i} (E_{f})

for all $i \geq 0$ and Template:Harv 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 Template:Harv or the introduction in Template:Harv).

References

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

@@ Line 1: / Line 1: @@
-== Abercrombie London  τολμώ να πω ==
+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]].
-Αγαπά πως υπάρχουν πάντα νέους συνεισφέροντες, νέα άρθρα για τις εργασίες και νέες αλλαγές που πρέπει να περιπολίες. Η μητέρα του, Melvine αγάπη, ήταν από την Τζαμάικα. Κανείς δεν αγοράζει The [http://www.eftihiadis.gr/templates/Finder.htm Abercrombie London] Darkness δεν πρόκειται να τους αγοράσετε, διότι η ίδια ετικέτα έχει υπογράψει GLC. Προσπαθήστε να μην διαβάζει πάρα πολύ σε ερωτήσεις. <br><br>Lopez γεννήθηκε 24 Ιούλη 1970 στη γειτονιά του Μπρονξ Castle Hill. «Είναι συνθήκες να είναι ανίκανοι και διεφθαρμένοι.". Συνεχίστε την ανάγνωση [Gramma Northe χαρά του Σκι με τα εγγόνια Φεβρουάριος 21, 2012There θα έρθει μια στιγμή στη ζωή του καθενός, όταν ο χρόνος θα αντιστραφεί. Εάν είναι δυνατόν, προσπαθήστε και να τους μιλήσω όταν φαίνονται αρκετά ανοικτή και όταν είστε τόσο στις καλές όσο moods.Don 't κάνει μια προσπάθεια η κάθε μία και κάθε φορά που θα πάρετε μαζί.. <br><br>Όλοι [http://www.eftihiadis.gr/templates/Finder.htm Abercrombie ελλαδα] γνωρίζαμε. Τα τελευταία χρόνια αυτό έχει γίνει εργάζονται στον τομέα της υγείας και της ασφάλειας, τολμώ να πω, μάλλον λαμπερό! Όλοι γνωρίζουμε για τα διαβατήρια για την υγεία και την ασφάλεια, αλλά και για μένα, είναι MD της εταιρίας εκπαίδευση για την υγεία και την [http://www.eftihiadis.gr/templates/Finder.htm Abercrombie And Fitch Greece] ασφάλεια μου έδωσε λόγο να χρησιμοποιούν τα παραδοσιακά μία μου πολλά ... <br><br>ITunes, cuz είναι ο ηγέτης στην αγορά αυτή γενικότερα, παίρνει για να φέρει αυτό στην αγορά για πρώτη φορά. Δεν κινείται με τη δική μου φιλοδοξία, Dean δήλωσε στο CNN σε συνέντευξή του στο συνέδριο Netroots Nation, μια ετήσια συνάντηση του αριστερού κλίνει πολιτικών ακτιβιστών. <br><br>Η φλαμανδική γλώσσα είναι συχνά αποκαλείται "Βέλγικα Ολλανδικά," κύρια γλώσσα σε μικρό έθνος του Βελγίου. Καλέστε μας σήμερα ή να κάνει μια έρευνα σε απευθείας σύνδεση για να μάθετε περισσότερα .. Κατά την εβδομάδα που οδηγεί μέχρι το Γενάρη με 1.700 κατοικίες ολοκληρωθεί, Fiddler της Creek έχει περίπου 4.300 κατοικίες που πρόκειται να κατασκευαστεί. Όταν κάνετε κράτηση μέσω ιστοσελίδας διακοπές ενοικίαση (όπου μπορείτε να κάνετε κράτηση απευθείας με τους ιδιοκτήτες) ελέγχουν εάν ο ενοικίασης προσφέρει υπηρεσία ενοικίασης καθεστώτος εγγυήσεων για την προστασία κατά της απάτης στο διαδίκτυο, θα πρέπει [http://www.eftihiadis.gr/templates/Finder.htm Abercrombie Greece] ένα ακίνητο κράτηση μέσω του site τους να αποδειχθεί δόλια.. <br><br>Είναι η πρώτη σας φορά, ώστε ό, τι πρέπει να ξέρετε είναι πώς να αγοράσει τα σωστά εργαλεία και πώς να αποφευχθεί η απάτη κατά την αγορά .. Ήταν πάντα δημοφιλής με τους φιλελεύθερους ακτιβιστές σε κανένα μικρό μέρος λόγω της fightin στάση του σώματος και την ειλικρίνεια του στην επικοινωνία φιλελεύθερες ιδέες που είναι έξω από το κύριο ρεύμα της αμερικανικής εκλογικού σώματος..<ul>
+==Motivation==
+Let <math>k</math> be a [[field (mathematics)|field]] and
-   <li>[http://galleri.kjerringoy.no/displayimage.php?album=155&pos=46 http://galleri.kjerringoy.no/displayimage.php?album=155&pos=46]</li>
+:<math>H_\ast(-)=H_\ast(-,k), H^\ast(-)=H^\ast(-,k)</math>
-   <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li>
+denote [[singular homology]] and [[singular cohomology]] with coefficients in ''k'', respectively.
-   <li>[http://enseignement-lsf.com/spip.php?article66#forum24001776 http://enseignement-lsf.com/spip.php?article66#forum24001776]</li>
+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>
-   <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
+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'' &times; ''X''. In this case the [[Künneth formula]] says
- </ul>
+:''H''<sup>&lowast;</sup>(''E<sub>f</sub>'') = ''H''<sup>&lowast;</sup>(''X''&times;''E'') &cong; ''H''<sup>&lowast;</sup>(''X'') ⊗<sub>k</sub> ''H''<sup>&lowast;</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&ndash;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'': ''&Delta;<sup>n</sup>'' → ''B'' the composition of ''s'' and the diagonal inclusion ''B'' ⊂ ''B'' &times; ''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>(&minus;), 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]]

Gaisser–Hillas function: Difference between revisions

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