Namespaces
Variants
Actions

Difference between revisions of "Spectral sequence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
<!--
 +
s0864901.png
 +
$#A+1 = 159 n = 0
 +
$#C+1 = 159 : ~/encyclopedia/old_files/data/S086/S.0806490 Spectral sequence
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).
 
A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).
  
All known spectral sequences can be obtained from exact couples. An exact couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864901.png" /> is defined as an exact diagram of the form
+
All known spectral sequences can be obtained from exact couples. An exact couple $  ( D  ^ {1} , E  ^ {1} , i  ^ {1} , j  ^ {1} , k  ^ {1} ) $
 +
is defined as an exact diagram of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864902.png" /></td> </tr></table>
+
$$
  
The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864903.png" /> is a differential in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864904.png" />. From any exact couple one can construct the derived exact couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864905.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864907.png" />. By iterating this construction one obtains the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864908.png" />.
+
\begin{array}{lcccl}
 +
D  ^ {1}  &{}  & \rightarrow ^ { {i  ^ {1}} }  &{}  &D  ^ {1}  \\
 +
{}  &{} _ {k  ^ {1}  }  &{}  &{} _ {j  ^ {1}  }  &{}  \\
 +
{}  &{}  &E  ^ {1}  &{}  &{}  \\
 +
\end{array}
  
1) The Leray spectral sequence. A filtered chain complex of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864909.png" /> determines an exact couple of bigraded modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649011.png" />. In the associated spectral sequence, the bidegree of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649012.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649013.png" />, and
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649014.png" /></td> </tr></table>
+
The homomorphism  $  d  ^ {1} = j  ^ {1} k  ^ {1} $
 +
is a differential in  $  E  ^ {1} $.
 +
From any exact couple one can construct the derived exact couple  $  ( D  ^ {2} , E  ^ {2} , i  ^ {2} , j  ^ {2} , k  ^ {2} ) $,
 +
for which  $  D  ^ {2} = \mathop{\rm Im}  i  ^ {1} $
 +
and  $  E  ^ {2} = H( E  ^ {1} , d  ^ {1} ) $.
 +
By iterating this construction one obtains the spectral sequence  $  E = \{ E  ^ {n} , d  ^ {n} \} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649015.png" /></td> </tr></table>
+
1) The Leray spectral sequence. A filtered chain complex of modules  $  ( \{ K  ^ {p} \} , d) $
 +
determines an exact couple of bigraded modules  $  D _ {p,q}  ^ {1} = H _ {p+} q ( K  ^ {p} ) $,
 +
$  E _ {p,q}  ^ {1} = H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 ) $.  
 +
In the associated spectral sequence, the bidegree of the differential  $  d  ^ {r} $
 +
is equal to  $  (- r, r- 1) $,
 +
and
  
The modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649016.png" /> form a filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649017.png" />. The bigraded module
+
$$
 +
E _ {p,q}  ^ {r}  =
 +
\frac{ \mathop{\rm Ker} ( d _ {p,q}  ^ {r-} 1 : E _ {p,q}  ^ {r-} 1 \rightarrow E _ {p-} r+ 1,q+ r- 2  ^ {r-} 1 ) }{
 +
\mathop{\rm Im} ( d _ {p+} r- 1,q- r+ 2  ^ {r-} 1 : E _ {p+} r- 1,q- r+ 2  ^ {r-} 1 \rightarrow E _ {p,q}  ^ {r-} 1 ) }
 +
  \simeq
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649018.png" /></td> </tr></table>
+
$$
 +
\simeq 
 +
\frac{ \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} / K  ^ {p-} r ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }{ \mathop{\rm Im} ( \partial
 +
: H _ {p+} q+ 1 ( K  ^ {p+} r- 1 / K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649019.png" /></td> </tr></table>
+
The modules  $  F _ {p,q} = \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} ) \rightarrow H _ {p+} q ( K)) $
 +
form a filtration of  $  H _ {*} ( K) $.  
 +
The bigraded module
  
is called the associated graded module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649020.png" />. The filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649021.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649022.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649024.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649026.png" />. For a regular filtration, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649027.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649028.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649029.png" />; such a spectral sequence is called a first-quadrant spectral sequence. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649030.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649031.png" />. In this case one says that the spectral sequence converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649032.png" />, and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649033.png" />.
+
$$
 +
E _ {p,q}  ^  \infty  = F _ {p,q} / F _ {p-} 1,q+ 1  \simeq
 +
$$
  
2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649034.png" /> by its skeletons gives the collapsing spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649035.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649036.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649038.png" />. The Leray–Serre spectral sequence is obtained from the filtration of the total space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649039.png" /> of the Serre fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649040.png" /> by the pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649041.png" /> of the skeletons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649042.png" /> of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649043.png" />. If the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649044.png" /> and base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649045.png" /> are path-connected, then for every coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649046.png" /> this gives the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649047.png" /> with differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649048.png" /> of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649049.png" /> for which
+
$$
 +
\simeq 
 +
\frac{ \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }{ \mathop{\rm Im} (
 +
\partial  : H _ {p+} q+ 1 ( K / K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649050.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649051.png" /> is a system of local coefficients over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649052.png" /> consisting of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649053.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649054.png" /> coincides with the composite
+
is called the associated graded module of  $  H _ {*} ( K) $.
 +
The filtration  $  \{ K  ^ {p} \} $
 +
is called regular if  $  K  ^ {p} = 0 $
 +
when  $  p< 0 $,
 +
$  E _ {p,q}  ^ {1} = 0 $
 +
when  $  q< 0 $
 +
and  $  K = \cup K  ^ {p} $.  
 +
For a regular filtration,  $  E _ {p,q}  ^ {r} = 0 $
 +
when  $  p< 0 $
 +
or  $  q< 0 $;
 +
such a spectral sequence is called a first-quadrant spectral sequence. Moreover,  $  E _ {p,q}  ^ {r} \simeq E _ {p,q}  ^ {r+} 1 \simeq E _ {p,q}  ^  \infty  $
 +
when  $  r > \max ( p, q+ 1) $.  
 +
In this case one says that the spectral sequence converges to  $  H _ {*} ( K) $,
 +
and writes  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( K) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649055.png" /></td> </tr></table>
+
2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a [[CW-complex|CW-complex]]  $  X $
 +
by its skeletons gives the collapsing spectral sequence  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( X) $,
 +
for which  $  E _ {p,q}  ^ {r} = E _ {p,q}  ^  \infty  = 0 $
 +
when  $  q \neq 0 $
 +
and  $  E _ {n,0}  ^ {2} = E _ {n,0}  ^  \infty  = H _ {n} ( X) $.  
 +
The Leray–Serre spectral sequence is obtained from the filtration of the total space  $  E $
 +
of the Serre fibration  $  F \rightarrow  ^ {i} E \rightarrow  ^ {p} B $
 +
by the pre-images  $  p  ^ {-} 1 ( B  ^ {n} ) $
 +
of the skeletons  $  B  ^ {n} $
 +
of the base  $  B $.  
 +
If the fibre  $  F $
 +
and base  $  B $
 +
are path-connected, then for every coefficient group  $  G $
 +
this gives the spectral sequence  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( E, G) $
 +
with differentials  $  d  ^ {r} $
 +
of bidegree  $  ( - r, r- 1) $
 +
for which
  
and the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649056.png" /> coincides with the composite
+
$$
 +
E _ {p,q}  ^ {1}  \simeq  C _ {p} ( B) \otimes H _ {q} ( F;  G)
 +
\  \textrm{ and } \  E _ {p,q}  ^ {2}  \simeq  H _ {p} ( B;  {\mathcal H} _ {q} ( F;  G)),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649057.png" /></td> </tr></table>
+
where  $  {\mathcal H} _ {q} ( F;  G) $
 +
is a system of local coefficients over  $  B $
 +
consisting of the groups  $  H _ {q} ( F; G) $.
 +
The homomorphism  $  i _ {*} : H _ {n} ( F; G) \rightarrow H _ {n} ( E; G) $
 +
coincides with the composite
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649058.png" /> is sufficiently large. The differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649059.png" /> of the spectral sequence coincides with the [[Transgression|transgression]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649060.png" />.
+
$$
 +
H _ {n} ( F;  G)  = E _ {0,n}  ^ {2}  \rightarrow  E _ {0,n}  ^ {r}
 +
= E _ {0,n}  ^  \infty  = F _ {0,n}  \subset  H _ {n} ( F;  G),
 +
$$
  
This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649061.png" />, with differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649062.png" /> of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649063.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649064.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649065.png" /> is a ring, then every term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649066.png" /> is a bigraded ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649067.png" /> is differentiation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649068.png" />, and the multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649069.png" /> is induced by that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649071.png" /> is a field and the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649072.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649073.png" />.
+
and the homomorphism  $  p _ {*} : H _ {n} ( E;  G) \rightarrow H _ {n} ( B;  G) $
 +
coincides with the composite
  
3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649074.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649075.png" />) to the same filtration of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649076.png" />. In its cohomological version, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649078.png" />. In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649079.png" /> is in general non-collapsing.
+
$$
 +
H _ {n} ( E;  G)  =  F _ {n,0}  \rightarrow  E _ {n,0}  ^  \infty  = \
 +
E _ {n,0}  ^ {r}  \subset  \
 +
E _ {n,0}  ^ {2}  =  H _ {n} ( B;  G),
 +
$$
 +
 
 +
where  $  r $
 +
is sufficiently large. The differential  $  d _ {n,0}  ^ {n} $
 +
of the spectral sequence coincides with the [[Transgression|transgression]]  $  \tau :  H _ {n} ( B;  G) \rightarrow H _ {n-} 1 ( F;  G) $.
 +
 
 +
This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence  $  E _ {r}  ^ {p,q} \Rightarrow H  ^ {p+} q ( E;  G) $,
 +
with differentials  $  d _ {r} $
 +
of bidegree  $  ( r, - r+ 1) $,
 +
for which  $  E _ {2}  ^ {p,q} \simeq H  ^ {p} ( B;  {\mathcal H} _ {q} ( F;  G)) $.
 +
If  $  G $
 +
is a ring, then every term  $  E _ {r} $
 +
is a bigraded ring,  $  d _ {r} $
 +
is differentiation in  $  E _ {r} $,
 +
and the multiplication in  $  E _ {r+} 1 $
 +
is induced by that in  $  E _ {r} $.
 +
If  $  G $
 +
is a field and the base  $  B $
 +
is simply connected, then  $  E _ {2}  ^ {**} \simeq H  ^ {*} ( B;  G) \otimes H  ^ {*} ( F;  G) $.
 +
 
 +
3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor $  h _ {*} $(
 +
$  h  ^ {*} $)  
 +
to the same filtration of the space $  E $.  
 +
In its cohomological version, $  E _ {r}  ^ {p,q} \Rightarrow h  ^ {p+} q ( E) $,
 +
$  E _ {2}  ^ {p,q} = H  ^ {p} ( B;  h  ^ {q} ( F  )) $.  
 +
In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration $  \mathop{\rm id} : X \rightarrow X $
 +
is in general non-collapsing.
  
 
4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations
 
4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649080.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{ccc}
 +
E  & \rightarrow  & X  \\
 +
\downarrow  &{}  &\downarrow  \\
 +
Y  & \rightarrow  & B  \\
 +
\end{array}
 +
 
 +
$$
  
 
In its cohomological version,
 
In its cohomological version,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649081.png" /></td> </tr></table>
+
$$
 +
E _ {r}  \Rightarrow  H  ^ {*} ( E; R),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649082.png" /></td> </tr></table>
+
$$
 +
E _ {2}  ^ {p,q}  \simeq  \mathop{\rm Tor} _ {H  ^ {*}
 +
( B;R) }  ^ {p,q} ( H  ^ {*} ( X; R); H  ^ {*} ( Y;  R)).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649083.png" /> is a field and the square consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649084.png" />-spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649085.png" />-mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.
+
If $  R $
 +
is a field and the square consists of $  H $-
 +
spaces and $  H $-
 +
mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.
  
5) The Adams spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649086.png" /> is defined for every prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649087.png" /> and all spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649089.png" /> (satisfying certain finiteness conditions). One has
+
5) The Adams spectral sequence $  E _ {r}  ^ {s,t} $
 +
is defined for every prime $  p\geq  2 $
 +
and all spaces $  X $
 +
and $  Y $(
 +
satisfying certain finiteness conditions). One has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649090.png" /></td> </tr></table>
+
$$
 +
E _ {2}  ^ {s,t}  \simeq  \mathop{\rm Ext} _ {A _ {p}  }  ^ {s,t} ( H
 +
^ {*} ( X; \mathbf Z _ {p} ); H  ^ {*} ( Y ; \mathbf Z _ {p} )),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649091.png" /> is the [[Steenrod algebra|Steenrod algebra]] modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649092.png" />. The bidegree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649093.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649094.png" />. This spectral sequence converges in the sense that, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649095.png" />, there is a monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649096.png" />, and so the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649097.png" /> is defined. There is a decreasing filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649098.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649099.png" /> of stable homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490100.png" /> such that
+
where $  A _ {p} $
 +
is the [[Steenrod algebra|Steenrod algebra]] modulo $  p $.  
 +
The bidegree of $  d _ {r} $
 +
is equal to $  ( r, r- 1) $.  
 +
This spectral sequence converges in the sense that, when $  r> s $,  
 +
there is a monomorphism $  E _ {r+} 1  ^ {s,t} \rightarrow E _ {r}  ^ {s,t} $,
 +
and so the group $  E _  \infty  ^ {s,t} = \cap _ {r>} s E _ {r}  ^ {s,t} $
 +
is defined. There is a decreasing filtration $  \{ F ^ { s } \} $
 +
of the group $  \{ Y, X \} $
 +
of stable homotopy classes of mappings $  Y \rightarrow X $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490101.png" /></td> </tr></table>
+
$$
 +
F ^ { s } \{ S  ^ {t-} s YX \} / F ^ { s+ 1 } \{ S  ^ {t-} s Y, X \}  \simeq  E _  \infty  ^ {s,t} ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490102.png" /> consists of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490103.png" /> of finite order prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490104.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490105.png" />, this spectral sequence enables one  "in principle"  to calculate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490106.png" />-components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary [[Generalized cohomology theories|generalized cohomology theories]]. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.
+
and $  F ^ { \infty } = \cap _ {s\geq } 0 F ^ { s } $
 +
consists of all elements of $  \{ Y, X \} $
 +
of finite order prime with $  p $.  
 +
When $  X= Y= S  ^ {0} $,  
 +
this spectral sequence enables one  "in principle"  to calculate the $  p $-
 +
components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary [[Generalized cohomology theories|generalized cohomology theories]]. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theory"  ''Math. USSR Izv.'' , '''31'''  (1967)  pp. 827–913  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L. Smith,  "Lectures on the Eilenberg–Moore spectral sequence" , ''Lect. notes in math.'' , '''134''' , Springer  (1970)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  D.C. Ravenel,  "A novices guide to the Adams–Novikov spectral sequence" , ''Geometric Applications of Homotopy Theory'' , '''2''' , Springer  (1978)  pp. 404–475</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theory"  ''Math. USSR Izv.'' , '''31'''  (1967)  pp. 827–913  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L. Smith,  "Lectures on the Eilenberg–Moore spectral sequence" , ''Lect. notes in math.'' , '''134''' , Springer  (1970)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  D.C. Ravenel,  "A novices guide to the Adams–Novikov spectral sequence" , ''Geometric Applications of Homotopy Theory'' , '''2''' , Springer  (1978)  pp. 404–475</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490108.png" /> be a spectral sequence, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490109.png" /> is the homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490110.png" />. A spectral sequence defines a series of modules of the initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490111.png" />, defined inductively as follows:
+
Let $  ( E  ^ {n} , d  ^ {n} ) $,
 +
$  n = 2, 3 \dots $
 +
be a spectral sequence, so that $  E  ^ {n+} 1 $
 +
is the homology of $  ( E  ^ {n} , d  ^ {n} ) $.  
 +
A spectral sequence defines a series of modules of the initial term $  E  ^ {2} $,  
 +
defined inductively as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490112.png" /></td> </tr></table>
+
$$
 +
0 = B  ^ {1}  \subset  B  ^ {2}  \subset  B  ^ {3}  \subset  \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490113.png" /></td> </tr></table>
+
$$
 +
\dots \subset  C  ^ {3}  \subset  C  ^ {2}  \subset  C  ^ {1}  = E  ^ {2} ,
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490114.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490115.png" /> is the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490116.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490117.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490118.png" />. One now defines the infinity terms:
+
with $  E  ^ {r+} 1 = C  ^ {r} / B  ^ {r} $,  
 +
and $  C  ^ {r+} 1 /B  ^ {r} $
 +
is the kernel of $  d  ^ {r} : E  ^ {r} \rightarrow E  ^ {r} $,  
 +
while $  B  ^ {r+} 1 /B  ^ {r} $
 +
is the image of $  d  ^ {r} $.  
 +
One now defines the infinity terms:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490119.png" /></td> </tr></table>
+
$$
 +
C  ^  \infty  = \cap _ { n } C  ^ {n} ,\  B  ^  \infty  = \
 +
\cup _ { n } B  ^ {n} ,\  E  ^  \infty  = C  ^  \infty  / B  ^  \infty  .
 +
$$
  
The terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490120.png" /> are thought of as successive approximations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490121.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490122.png" /> is a spectral sequence of bigraded modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490124.png" />, all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490129.png" /> also carry corresponding natural bigraded structures.
+
The terms $  E  ^ {r} $
 +
are thought of as successive approximations of $  E  ^  \infty  $.  
 +
If $  ( E  ^ {n} , d  ^ {n} ) $
 +
is a spectral sequence of bigraded modules $  E  ^ {n} = \oplus E _ {p,q}  ^ {n} $,
 +
$  d  ^ {r} : E _ {p,q}  ^ {r} \rightarrow E _ {p-} r,q+ r- 1  ^ {r} $,  
 +
all the $  B  ^ {i} $,  
 +
$  C  ^ {i} $,  
 +
$  B  ^  \infty  $,  
 +
$  C  ^  \infty  $,  
 +
$  E  ^  \infty  $
 +
also carry corresponding natural bigraded structures.
  
Sometimes there is an initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490130.png" />, and then the same construction is carried out with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490131.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490132.png" />.
+
Sometimes there is an initial term $  E  ^ {1} $,  
 +
and then the same construction is carried out with $  E  ^ {1} $
 +
instead of $  E  ^ {2} $.
  
For a first-quadrant spectral sequence, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490133.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490134.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490135.png" />, for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490136.png" /> and large enough <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490137.png" /> one has that in
+
For a first-quadrant spectral sequence, i.e. $  E _ {p,q}  ^ {2} = 0 $
 +
for $  p< 0 $
 +
or $  q< 0 $,  
 +
for given $  p, q $
 +
and large enough $  r $
 +
one has that in
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490138.png" /></td> </tr></table>
+
$$
 +
E _ {p+} r,q- r+ 1  ^ {r}  \rightarrow ^ { {d  ^ {r}} }  E _ {p,q}  ^ {r}
 +
\rightarrow ^ { {d  ^ {r}} }  E _ {p-} r,q+ r- 1  ^ {r}
 +
$$
  
both the outside modules are zero, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490139.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490140.png" /> large enough.
+
both the outside modules are zero, so that $  E _ {p,q}  ^ {r} = E _ {p,q}  ^ {r+} 1 = E _ {p,q}  ^  \infty  $
 +
for $  r $
 +
large enough.
  
For a first-quadrant spectral sequence one also always has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490141.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490142.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490143.png" /> is a quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490144.png" />, giving rise to sequences of monomorphisms and epimorphisms:
+
For a first-quadrant spectral sequence one also always has that $  E _ {p,0}  ^ {r+} 1 $
 +
is a submodule of $  E _ {p,0}  ^ {r} $,  
 +
and $  E _ {0,q}  ^ {r+} 1 $
 +
is a quotient of $  E _ {0,q}  ^ {r} $,
 +
giving rise to sequences of monomorphisms and epimorphisms:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490145.png" /></td> </tr></table>
+
$$
 +
E _ {p,0}  ^  \infty  = E _ {p,0}  ^ {p+} 1  \rightarrow \dots \rightarrow  E _ {p,0}  ^ {3}  \rightarrow  E _ {p,0}  ^ {2} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490146.png" /></td> </tr></table>
+
$$
 +
E _ {0,q}  ^ {2}  \rightarrow  E _ {0,q}  ^ {3}  \rightarrow
 +
\dots \rightarrow  E _ {0,q}  ^ {q+} 2  = E _ {0,q}  ^  \infty  ,
 +
$$
  
 
which are known as the edge homomorphisms.
 
which are known as the edge homomorphisms.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490147.png" /> be a filtration of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490148.png" /> by submodules
+
Let $  ( A _ {p} ) $
 +
be a filtration of a module $  A $
 +
by submodules
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490149.png" /></td> </tr></table>
+
$$
 +
\dots \subset  A _ {p-} 1  \subset  A _ {p}  \subset  A _ {p+} 1  \subset  \dots
 +
$$
  
with associated graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490150.png" />:
+
with associated graded module $  \mathop{\rm Gr} ( A) $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490151.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Gr} ( A)  = \oplus _ { p } A _ {p} / A _ {p-} 1 .
 +
$$
  
A spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490152.png" /> is said to converge to a graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490153.png" />, in symbols
+
A spectral sequence $  ( E _ {p}  ^ {r} , d  ^ {r} ) $
 +
is said to converge to a graded module $  H $,  
 +
in symbols
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490154.png" /></td> </tr></table>
+
$$
 +
E _ {p}  ^ {r}  \Rightarrow  H ,
 +
$$
  
if there is a filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490155.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490156.png" /> such that
+
if there is a filtration $  F _ {p} H $
 +
of $  H $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490157.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
E _ {p}  ^  \infty  \simeq  F _ {p} H /F _ {p+} 1 H .
 +
$$
  
In the usual cases the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490159.png" /> are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.
+
In the usual cases the $  E _ {p}  ^ {r} $
 +
and $  H $
 +
are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.

Latest revision as of 14:55, 7 June 2020


A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary Abelian category (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).

All known spectral sequences can be obtained from exact couples. An exact couple $ ( D ^ {1} , E ^ {1} , i ^ {1} , j ^ {1} , k ^ {1} ) $ is defined as an exact diagram of the form

$$ \begin{array}{lcccl} D ^ {1} &{} & \rightarrow ^ { {i ^ {1}} } &{} &D ^ {1} \\ {} &{} _ {k ^ {1} } &{} &{} _ {j ^ {1} } &{} \\ {} &{} &E ^ {1} &{} &{} \\ \end{array} $$

The homomorphism $ d ^ {1} = j ^ {1} k ^ {1} $ is a differential in $ E ^ {1} $. From any exact couple one can construct the derived exact couple $ ( D ^ {2} , E ^ {2} , i ^ {2} , j ^ {2} , k ^ {2} ) $, for which $ D ^ {2} = \mathop{\rm Im} i ^ {1} $ and $ E ^ {2} = H( E ^ {1} , d ^ {1} ) $. By iterating this construction one obtains the spectral sequence $ E = \{ E ^ {n} , d ^ {n} \} $.

1) The Leray spectral sequence. A filtered chain complex of modules $ ( \{ K ^ {p} \} , d) $ determines an exact couple of bigraded modules $ D _ {p,q} ^ {1} = H _ {p+} q ( K ^ {p} ) $, $ E _ {p,q} ^ {1} = H _ {p+} q ( K ^ {p} / K ^ {p-} 1 ) $. In the associated spectral sequence, the bidegree of the differential $ d ^ {r} $ is equal to $ (- r, r- 1) $, and

$$ E _ {p,q} ^ {r} = \frac{ \mathop{\rm Ker} ( d _ {p,q} ^ {r-} 1 : E _ {p,q} ^ {r-} 1 \rightarrow E _ {p-} r+ 1,q+ r- 2 ^ {r-} 1 ) }{ \mathop{\rm Im} ( d _ {p+} r- 1,q- r+ 2 ^ {r-} 1 : E _ {p+} r- 1,q- r+ 2 ^ {r-} 1 \rightarrow E _ {p,q} ^ {r-} 1 ) } \simeq $$

$$ \simeq \frac{ \mathop{\rm Im} ( H _ {p+} q ( K ^ {p} / K ^ {p-} r ) \rightarrow H _ {p+} q ( K ^ {p} / K ^ {p-} 1 )) }{ \mathop{\rm Im} ( \partial : H _ {p+} q+ 1 ( K ^ {p+} r- 1 / K ^ {p} ) \rightarrow H _ {p+} q ( K ^ {p} / K ^ {p-} 1 )) } . $$

The modules $ F _ {p,q} = \mathop{\rm Im} ( H _ {p+} q ( K ^ {p} ) \rightarrow H _ {p+} q ( K)) $ form a filtration of $ H _ {*} ( K) $. The bigraded module

$$ E _ {p,q} ^ \infty = F _ {p,q} / F _ {p-} 1,q+ 1 \simeq $$

$$ \simeq \frac{ \mathop{\rm Im} ( H _ {p+} q ( K ^ {p} ) \rightarrow H _ {p+} q ( K ^ {p} / K ^ {p-} 1 )) }{ \mathop{\rm Im} ( \partial : H _ {p+} q+ 1 ( K / K ^ {p} ) \rightarrow H _ {p+} q ( K ^ {p} / K ^ {p-} 1 )) } $$

is called the associated graded module of $ H _ {*} ( K) $. The filtration $ \{ K ^ {p} \} $ is called regular if $ K ^ {p} = 0 $ when $ p< 0 $, $ E _ {p,q} ^ {1} = 0 $ when $ q< 0 $ and $ K = \cup K ^ {p} $. For a regular filtration, $ E _ {p,q} ^ {r} = 0 $ when $ p< 0 $ or $ q< 0 $; such a spectral sequence is called a first-quadrant spectral sequence. Moreover, $ E _ {p,q} ^ {r} \simeq E _ {p,q} ^ {r+} 1 \simeq E _ {p,q} ^ \infty $ when $ r > \max ( p, q+ 1) $. In this case one says that the spectral sequence converges to $ H _ {*} ( K) $, and writes $ E _ {p,q} ^ {r} \Rightarrow H _ {p+} q ( K) $.

2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a CW-complex $ X $ by its skeletons gives the collapsing spectral sequence $ E _ {p,q} ^ {r} \Rightarrow H _ {p+} q ( X) $, for which $ E _ {p,q} ^ {r} = E _ {p,q} ^ \infty = 0 $ when $ q \neq 0 $ and $ E _ {n,0} ^ {2} = E _ {n,0} ^ \infty = H _ {n} ( X) $. The Leray–Serre spectral sequence is obtained from the filtration of the total space $ E $ of the Serre fibration $ F \rightarrow ^ {i} E \rightarrow ^ {p} B $ by the pre-images $ p ^ {-} 1 ( B ^ {n} ) $ of the skeletons $ B ^ {n} $ of the base $ B $. If the fibre $ F $ and base $ B $ are path-connected, then for every coefficient group $ G $ this gives the spectral sequence $ E _ {p,q} ^ {r} \Rightarrow H _ {p+} q ( E, G) $ with differentials $ d ^ {r} $ of bidegree $ ( - r, r- 1) $ for which

$$ E _ {p,q} ^ {1} \simeq C _ {p} ( B) \otimes H _ {q} ( F; G) \ \textrm{ and } \ E _ {p,q} ^ {2} \simeq H _ {p} ( B; {\mathcal H} _ {q} ( F; G)), $$

where $ {\mathcal H} _ {q} ( F; G) $ is a system of local coefficients over $ B $ consisting of the groups $ H _ {q} ( F; G) $. The homomorphism $ i _ {*} : H _ {n} ( F; G) \rightarrow H _ {n} ( E; G) $ coincides with the composite

$$ H _ {n} ( F; G) = E _ {0,n} ^ {2} \rightarrow E _ {0,n} ^ {r} = E _ {0,n} ^ \infty = F _ {0,n} \subset H _ {n} ( F; G), $$

and the homomorphism $ p _ {*} : H _ {n} ( E; G) \rightarrow H _ {n} ( B; G) $ coincides with the composite

$$ H _ {n} ( E; G) = F _ {n,0} \rightarrow E _ {n,0} ^ \infty = \ E _ {n,0} ^ {r} \subset \ E _ {n,0} ^ {2} = H _ {n} ( B; G), $$

where $ r $ is sufficiently large. The differential $ d _ {n,0} ^ {n} $ of the spectral sequence coincides with the transgression $ \tau : H _ {n} ( B; G) \rightarrow H _ {n-} 1 ( F; G) $.

This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence $ E _ {r} ^ {p,q} \Rightarrow H ^ {p+} q ( E; G) $, with differentials $ d _ {r} $ of bidegree $ ( r, - r+ 1) $, for which $ E _ {2} ^ {p,q} \simeq H ^ {p} ( B; {\mathcal H} _ {q} ( F; G)) $. If $ G $ is a ring, then every term $ E _ {r} $ is a bigraded ring, $ d _ {r} $ is differentiation in $ E _ {r} $, and the multiplication in $ E _ {r+} 1 $ is induced by that in $ E _ {r} $. If $ G $ is a field and the base $ B $ is simply connected, then $ E _ {2} ^ {**} \simeq H ^ {*} ( B; G) \otimes H ^ {*} ( F; G) $.

3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor $ h _ {*} $( $ h ^ {*} $) to the same filtration of the space $ E $. In its cohomological version, $ E _ {r} ^ {p,q} \Rightarrow h ^ {p+} q ( E) $, $ E _ {2} ^ {p,q} = H ^ {p} ( B; h ^ {q} ( F )) $. In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration $ \mathop{\rm id} : X \rightarrow X $ is in general non-collapsing.

4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations

$$ \begin{array}{ccc} E & \rightarrow & X \\ \downarrow &{} &\downarrow \\ Y & \rightarrow & B \\ \end{array} $$

In its cohomological version,

$$ E _ {r} \Rightarrow H ^ {*} ( E; R), $$

$$ E _ {2} ^ {p,q} \simeq \mathop{\rm Tor} _ {H ^ {*} ( B;R) } ^ {p,q} ( H ^ {*} ( X; R); H ^ {*} ( Y; R)). $$

If $ R $ is a field and the square consists of $ H $- spaces and $ H $- mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.

5) The Adams spectral sequence $ E _ {r} ^ {s,t} $ is defined for every prime $ p\geq 2 $ and all spaces $ X $ and $ Y $( satisfying certain finiteness conditions). One has

$$ E _ {2} ^ {s,t} \simeq \mathop{\rm Ext} _ {A _ {p} } ^ {s,t} ( H ^ {*} ( X; \mathbf Z _ {p} ); H ^ {*} ( Y ; \mathbf Z _ {p} )), $$

where $ A _ {p} $ is the Steenrod algebra modulo $ p $. The bidegree of $ d _ {r} $ is equal to $ ( r, r- 1) $. This spectral sequence converges in the sense that, when $ r> s $, there is a monomorphism $ E _ {r+} 1 ^ {s,t} \rightarrow E _ {r} ^ {s,t} $, and so the group $ E _ \infty ^ {s,t} = \cap _ {r>} s E _ {r} ^ {s,t} $ is defined. There is a decreasing filtration $ \{ F ^ { s } \} $ of the group $ \{ Y, X \} $ of stable homotopy classes of mappings $ Y \rightarrow X $ such that

$$ F ^ { s } \{ S ^ {t-} s YX \} / F ^ { s+ 1 } \{ S ^ {t-} s Y, X \} \simeq E _ \infty ^ {s,t} , $$

and $ F ^ { \infty } = \cap _ {s\geq } 0 F ^ { s } $ consists of all elements of $ \{ Y, X \} $ of finite order prime with $ p $. When $ X= Y= S ^ {0} $, this spectral sequence enables one "in principle" to calculate the $ p $- components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary generalized cohomology theories. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.

References

[1] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[2] D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[3] J.-P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. , 54 (1951) pp. 425–505
[4] S. MacLane, "Homology" , Springer (1963)
[5] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[6] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[7] S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[8] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[9] S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theory" Math. USSR Izv. , 31 (1967) pp. 827–913 Izv. Akad. Nauk. SSSR Ser. Mat. , 31 (1967) pp. 855–951
[10] J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974)
[11] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[12] L. Smith, "Lectures on the Eilenberg–Moore spectral sequence" , Lect. notes in math. , 134 , Springer (1970)
[13] D.C. Ravenel, "A novices guide to the Adams–Novikov spectral sequence" , Geometric Applications of Homotopy Theory , 2 , Springer (1978) pp. 404–475

Comments

Let $ ( E ^ {n} , d ^ {n} ) $, $ n = 2, 3 \dots $ be a spectral sequence, so that $ E ^ {n+} 1 $ is the homology of $ ( E ^ {n} , d ^ {n} ) $. A spectral sequence defines a series of modules of the initial term $ E ^ {2} $, defined inductively as follows:

$$ 0 = B ^ {1} \subset B ^ {2} \subset B ^ {3} \subset \dots $$

$$ \dots \subset C ^ {3} \subset C ^ {2} \subset C ^ {1} = E ^ {2} , $$

with $ E ^ {r+} 1 = C ^ {r} / B ^ {r} $, and $ C ^ {r+} 1 /B ^ {r} $ is the kernel of $ d ^ {r} : E ^ {r} \rightarrow E ^ {r} $, while $ B ^ {r+} 1 /B ^ {r} $ is the image of $ d ^ {r} $. One now defines the infinity terms:

$$ C ^ \infty = \cap _ { n } C ^ {n} ,\ B ^ \infty = \ \cup _ { n } B ^ {n} ,\ E ^ \infty = C ^ \infty / B ^ \infty . $$

The terms $ E ^ {r} $ are thought of as successive approximations of $ E ^ \infty $. If $ ( E ^ {n} , d ^ {n} ) $ is a spectral sequence of bigraded modules $ E ^ {n} = \oplus E _ {p,q} ^ {n} $, $ d ^ {r} : E _ {p,q} ^ {r} \rightarrow E _ {p-} r,q+ r- 1 ^ {r} $, all the $ B ^ {i} $, $ C ^ {i} $, $ B ^ \infty $, $ C ^ \infty $, $ E ^ \infty $ also carry corresponding natural bigraded structures.

Sometimes there is an initial term $ E ^ {1} $, and then the same construction is carried out with $ E ^ {1} $ instead of $ E ^ {2} $.

For a first-quadrant spectral sequence, i.e. $ E _ {p,q} ^ {2} = 0 $ for $ p< 0 $ or $ q< 0 $, for given $ p, q $ and large enough $ r $ one has that in

$$ E _ {p+} r,q- r+ 1 ^ {r} \rightarrow ^ { {d ^ {r}} } E _ {p,q} ^ {r} \rightarrow ^ { {d ^ {r}} } E _ {p-} r,q+ r- 1 ^ {r} $$

both the outside modules are zero, so that $ E _ {p,q} ^ {r} = E _ {p,q} ^ {r+} 1 = E _ {p,q} ^ \infty $ for $ r $ large enough.

For a first-quadrant spectral sequence one also always has that $ E _ {p,0} ^ {r+} 1 $ is a submodule of $ E _ {p,0} ^ {r} $, and $ E _ {0,q} ^ {r+} 1 $ is a quotient of $ E _ {0,q} ^ {r} $, giving rise to sequences of monomorphisms and epimorphisms:

$$ E _ {p,0} ^ \infty = E _ {p,0} ^ {p+} 1 \rightarrow \dots \rightarrow E _ {p,0} ^ {3} \rightarrow E _ {p,0} ^ {2} , $$

$$ E _ {0,q} ^ {2} \rightarrow E _ {0,q} ^ {3} \rightarrow \dots \rightarrow E _ {0,q} ^ {q+} 2 = E _ {0,q} ^ \infty , $$

which are known as the edge homomorphisms.

Let $ ( A _ {p} ) $ be a filtration of a module $ A $ by submodules

$$ \dots \subset A _ {p-} 1 \subset A _ {p} \subset A _ {p+} 1 \subset \dots $$

with associated graded module $ \mathop{\rm Gr} ( A) $:

$$ \mathop{\rm Gr} ( A) = \oplus _ { p } A _ {p} / A _ {p-} 1 . $$

A spectral sequence $ ( E _ {p} ^ {r} , d ^ {r} ) $ is said to converge to a graded module $ H $, in symbols

$$ E _ {p} ^ {r} \Rightarrow H , $$

if there is a filtration $ F _ {p} H $ of $ H $ such that

$$ \tag{* } E _ {p} ^ \infty \simeq F _ {p} H /F _ {p+} 1 H . $$

In the usual cases the $ E _ {p} ^ {r} $ and $ H $ are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.

How to Cite This Entry:
Spectral sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_sequence&oldid=12982
This article was adapted from an original article by S.N. Malygin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article