Namespaces
Variants
Actions

Difference between revisions of "Steenrod square"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A stable [[Cohomology operation|cohomology operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875602.png" />, of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875603.png" />, which raises the dimension by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875604.png" />. This means that for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875605.png" /> and every pair of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875606.png" />, a homomorphism
+
<!--
 +
s0875601.png
 +
$#A+1 = 62 n = 0
 +
$#C+1 = 62 : ~/encyclopedia/old_files/data/S087/S.0807560 Steenrod square
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/s087/s087560/s0875607.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
is defined such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875608.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s0875609.png" /> is the coboundary homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756010.png" /> (stability) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756011.png" /> for any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756012.png" /> (naturality). The Steenrod squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756013.png" /> possess the following properties:
+
A stable [[Cohomology operation|cohomology operation]]  $  Sq  ^ {i} $,
 +
$  i \geq  0 $,  
 +
of the type  $  ( \mathbf Z _ {2} , \mathbf Z _ {2} ) $,
 +
which raises the dimension by  $  i $.  
 +
This means that for every integer  $  n $
 +
and every pair of topological spaces  $  ( X, Y) $,
 +
a homomorphism
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756014.png" />;
+
$$
 +
Sq  ^ {i} :  H  ^ {n} ( X, Y;  \mathbf Z _ {2} ) \rightarrow  H  ^ {n+} i ( X, Y; \mathbf Z _ {2} )
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756016.png" /> is the Bockstein homomorphism associated with the short exact sequence of coefficient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756017.png" />;
+
is defined such that  $  \delta Sq  ^ {i} = Sq  ^ {i} \delta $,  
 +
where $  \delta $
 +
is the coboundary homomorphism $  \delta :  H  ^ {q} ( Y;  \mathbf Z _ {2} ) \rightarrow H  ^ {q+} 1 ( X, Y;  \mathbf Z _ {2} ) $(
 +
stability) and  $  f ^ { \star } Sq  ^ {i} = Sq  ^ {i} f ^ { \star } $
 +
for any continuous mapping  $  f: ( X, Y) \rightarrow ( X  ^  \prime  , Y  ^  \prime  ) $(
 +
naturality). The Steenrod squares  $  Sq  ^ {i} $
 +
possess the following properties:
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756019.png" />;
+
1) $  Sq  ^ {0} = \mathop{\rm id} $;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756021.png" />;
+
2) $  Sq  ^ {1} = \beta $,
 +
where  $  \beta $
 +
is the Bockstein homomorphism associated with the short exact sequence of coefficient groups  $  0 \rightarrow \mathbf Z _ {2} \rightarrow \mathbf Z _ {4} \rightarrow \mathbf Z _ {2} \rightarrow 0 $;
  
5) (Cartan's formula) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756022.png" />;
+
3) if  $  i = \mathop{\rm dim}  x $,
 +
then  $  Sq  ^ {i} x = x  ^ {2} $;
  
6) (Adem relation) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756023.png" />, then
+
4) if $  i > \mathop{\rm dim}  x $,  
 +
then $  Sq  ^ {i} x = 0 $;
  
<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/s087/s087560/s08756024.png" /></td> </tr></table>
+
5) (Cartan's formula)  $  Sq  ^ {i} ( xy) = \sum _ {j=} 0  ^ {i} ( Sq  ^ {i} x) \cdot ( Sq  ^ {i-} 1 y) $;
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756025.png" /> are binomial coefficients modulo 2.
+
6) (Adem relation) if  $  a < 2b $,
 +
then
  
In Cartan's formula, multiplication can be considered as outer (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756026.png" />-multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756027.png" />, defined by the formula
+
$$
 +
Sq  ^ {a} Sq  ^ {b}  = \sum _ { t= } 0 ^ { [ }  a/2] \left ( \begin{array}{c}
 +
b- t- 1 \\
 +
a- 2t
 +
\end{array}
 +
\right ) _ {2} Sq  ^ {a+} b- t Sq  ^ {t} ,
 +
$$
  
<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/s087/s087560/s08756028.png" /></td> </tr></table>
+
where  $  ( \cdot ) _ {2} $
 +
are binomial coefficients modulo 2.
  
is a ring homomorphism. It follows from the stability condition that the Steenrod squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756029.png" /> commute with [[Suspension|suspension]] and [[Transgression|transgression]].
+
In Cartan's formula, multiplication can be considered as outer ( $  \times $-
 +
multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping  $  Sq: H  ^  \star  ( X;  \mathbf Z _ {2} ) \rightarrow H  ^  \star  ( X;  \mathbf Z _ {2} ) $,
 +
defined by the formula
  
The properties 1), 3) and 4) define the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756030.png" /> uniquely and can therefore be taken as defining axioms. The constructive definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756031.png" /> is based on the simplicial structure in chain groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756032.png" /> and on the existence of a diagonal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756033.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756034.png" /> be the minimal acyclic free chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756035.png" />-complex i.e. the chain complex for which
+
$$
 +
Sqx  =  x + Sq  ^ {1} x + \dots + Sq  ^ {n-} 1 x + x  ^ {2} ,\ \
 +
x \in H  ^ {n} ( X;  \mathbf Z _ {2} ),
 +
$$
  
<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/s087/s087560/s08756036.png" /></td> </tr></table>
+
is a ring homomorphism. It follows from the stability condition that the Steenrod squares  $  Sq  ^ {i} $
 +
commute with [[Suspension|suspension]] and [[Transgression|transgression]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756037.png" /> is the generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756038.png" />. The existence of an equivariant chain mapping
+
The properties 1), 3) and 4) define the operations  $  Sq  ^ {i} $
 +
uniquely and can therefore be taken as defining axioms. The constructive definition of  $  Sq  ^ {i} $
 +
is based on the simplicial structure in chain groups  $  C _  \star  ( X) $
 +
and on the existence of a diagonal mapping  $  \Delta : X \rightarrow X \times X $.  
 +
Let  $  W $
 +
be the minimal acyclic free chain  $  \mathbf Z _ {2} $-
 +
complex i.e. the chain complex for which
  
<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/s087/s087560/s08756039.png" /></td> </tr></table>
+
$$
 +
W _ {i}  = \mathbf Z _ {2} [ e _ {i} , Te _ {i} ] ,\ \
 +
de _ {i}  = e _ {i-} 1 + (- 1)  ^ {i} Te _ {i-} 1 ,
 +
$$
 +
 
 +
where  $  T $
 +
is the generator of  $  \mathbf Z _ {2} $.
 +
The existence of an equivariant chain mapping
 +
 
 +
$$
 +
\phi : W \otimes C _  \star  ( X)  \rightarrow  C _  \star  ( X) \otimes C _  \star  ( X)
 +
$$
  
 
such that
 
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/s087/s087560/s08756040.png" /></td> </tr></table>
+
$$
 +
\phi : ( e _ {i} \otimes \sigma )  \in  C _  \star  ( \sigma \otimes \sigma )  \subset  C _  \star  ( X) \otimes C _  \star  ( X)  = C _  \star  ( X \times X)
 +
$$
  
for any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756041.png" />, is proved by the method of acyclic carriers or by an explicit construction (see [[#References|[4]]]). The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756042.png" /> here signifies the smallest subcomplex of the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756043.png" /> containing the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756045.png" />. Any two cochains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756047.png" /> are put in correspondence by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756048.png" />, for any simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756049.png" />, with the cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756050.png" />, which is called their cup-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756052.png" />-product. For the coboundary of this chain, the formula
+
for any simplex $  \sigma \in C _  \star  ( X) $,  
 +
is proved by the method of acyclic carriers or by an explicit construction (see [[#References|[4]]]). The symbol $  C _  \star  ( \sigma \otimes \sigma ) $
 +
here signifies the smallest subcomplex of the chain complex $  C _  \star  ( X) \otimes C _  \star  ( X) $
 +
containing the element $  \sigma \otimes \sigma $.  
 +
Let $  i \geq  0 $.  
 +
Any two cochains $  u \in C  ^ {p} ( X) $,  
 +
$  v \in C  ^ {q} ( X) $
 +
are put in correspondence by the formula $  ( u \cup _ {i} v)( \sigma ) = ( u \otimes v)( \phi ( e _ {i} \otimes \sigma )) $,  
 +
for any simplex $  \sigma \in C _ {p+} q- i ( X) $,  
 +
with the cochain $  u \cup _ {i} v \in C  ^ {p+} q- i ( X) $,  
 +
which is called their cup- $  i $-
 +
product. For the coboundary of this chain, the formula
  
<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/s087/s087560/s08756053.png" /></td> </tr></table>
+
$$
 +
\delta \left ( u \cup _ { i } v \right )  = \
 +
(- 1)  ^ {i} \delta u \cup _ { i } v + (- 1)  ^ {i+} p u \cup _ { i } \delta v +
 +
$$
  
<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/s087/s087560/s08756054.png" /></td> </tr></table>
+
$$
 +
+
 +
(- 1)  ^ {i+} 1 u \cup _ { i- } 1 v + (- 1)  ^ {pq+} 1 v \cup _ { i- } 1 u
 +
$$
  
holds, from which it follows that the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756055.png" /> correctly defines a homomorphism
+
holds, from which it follows that the formula $  Sq  ^ {p-} i \{ u \} = \{ u \cup _ {i} u \} $
 +
correctly defines a homomorphism
  
<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/s087/s087560/s08756056.png" /></td> </tr></table>
+
$$
 +
Sq  ^ {p-} i : H  ^ {p} ( X; \mathbf Z _ {2} )  \rightarrow  H  ^ {2p-} i ( X; \mathbf Z _ {2} )
 +
$$
  
which does not depend on the choice of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756057.png" />.
+
which does not depend on the choice of the mapping $  \phi $.
  
The operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756058.png" /> are constructed in the same way in other simplicial structures with a diagonal mapping, for example, in cohomology groups of simplicial Abelian groups, of simplicial Lie algebras, etc. However, not all properties of the Steenrod squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756059.png" /> are preserved then (for example, generally speaking, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756060.png" />) and there is yet (1991) no single general theory for the generalized operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756061.png" /> (see [[#References|[5]]], [[#References|[6]]]).
+
The operations $  Sq  ^ {i} $
 +
are constructed in the same way in other simplicial structures with a diagonal mapping, for example, in cohomology groups of simplicial Abelian groups, of simplicial Lie algebras, etc. However, not all properties of the Steenrod squares $  Sq  ^ {i} $
 +
are preserved then (for example, generally speaking, $  Sq  ^ {0} \neq  \mathop{\rm id} $)  
 +
and there is yet (1991) no single general theory for the generalized operations $  Sq  ^ {i} $(
 +
see [[#References|[5]]], [[#References|[6]]]).
  
Many cohomology operations which act on cohomology groups with coefficients in the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087560/s08756063.png" /> can be expressed in terms of the Steenrod squares and their analogues (see [[Steenrod reduced power|Steenrod reduced power]]). This underlines the fundamental role played by Steenrod squares in algebraic topology and its applications. For example, [[Bordism|bordism]] groups are calculated using Steenrod squares.
+
Many cohomology operations which act on cohomology groups with coefficients in the groups $  \mathbf Z _ {2} $
 +
and $  \mathbf Z _ {p} $
 +
can be expressed in terms of the Steenrod squares and their analogues (see [[Steenrod reduced power|Steenrod reduced power]]). This underlines the fundamental role played by Steenrod squares in algebraic topology and its applications. For example, [[Bordism|bordism]] groups are calculated using Steenrod squares.
  
 
Steenrod squares were introduced by N. Steenrod [[#References|[4]]].
 
Steenrod squares were introduced by N. Steenrod [[#References|[4]]].
Line 59: Line 144:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations" , Princeton Univ. Press  (1962)</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">  M.K. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Steenrod,  "Products of cocylces and extensions of mappings"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 290–320</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Epstein,  "Steenrod operations in homological algebra"  ''Invent. Math.'' , '''1''' :  2  (1966)  pp. 152–208</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. May,  "A general algebraic approach to Steenrod operations" , ''The Steenrod Algebra and Its Applications'' , ''Lect. notes in math.'' , '''168''' , Springer  (1970)  pp. 153–231</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.E. Steenrod,  D.B.A. Epstein,  "Cohomology operations" , Princeton Univ. Press  (1962)</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">  M.K. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Steenrod,  "Products of cocylces and extensions of mappings"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 290–320</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Epstein,  "Steenrod operations in homological algebra"  ''Invent. Math.'' , '''1''' :  2  (1966)  pp. 152–208</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. May,  "A general algebraic approach to Steenrod operations" , ''The Steenrod Algebra and Its Applications'' , ''Lect. notes in math.'' , '''168''' , Springer  (1970)  pp. 153–231</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  ''Matematika'' , '''5''' :  2  (1961)  pp. 3–11; 11–30; 30–49; 50–102</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapt. 12</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. V, Sect. 9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,  "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. Chapt. 18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)  pp. Part III, Chapt. 12</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. V, Sect. 9</TD></TR></table>

Latest revision as of 08:23, 6 June 2020


A stable cohomology operation $ Sq ^ {i} $, $ i \geq 0 $, of the type $ ( \mathbf Z _ {2} , \mathbf Z _ {2} ) $, which raises the dimension by $ i $. This means that for every integer $ n $ and every pair of topological spaces $ ( X, Y) $, a homomorphism

$$ Sq ^ {i} : H ^ {n} ( X, Y; \mathbf Z _ {2} ) \rightarrow H ^ {n+} i ( X, Y; \mathbf Z _ {2} ) $$

is defined such that $ \delta Sq ^ {i} = Sq ^ {i} \delta $, where $ \delta $ is the coboundary homomorphism $ \delta : H ^ {q} ( Y; \mathbf Z _ {2} ) \rightarrow H ^ {q+} 1 ( X, Y; \mathbf Z _ {2} ) $( stability) and $ f ^ { \star } Sq ^ {i} = Sq ^ {i} f ^ { \star } $ for any continuous mapping $ f: ( X, Y) \rightarrow ( X ^ \prime , Y ^ \prime ) $( naturality). The Steenrod squares $ Sq ^ {i} $ possess the following properties:

1) $ Sq ^ {0} = \mathop{\rm id} $;

2) $ Sq ^ {1} = \beta $, where $ \beta $ is the Bockstein homomorphism associated with the short exact sequence of coefficient groups $ 0 \rightarrow \mathbf Z _ {2} \rightarrow \mathbf Z _ {4} \rightarrow \mathbf Z _ {2} \rightarrow 0 $;

3) if $ i = \mathop{\rm dim} x $, then $ Sq ^ {i} x = x ^ {2} $;

4) if $ i > \mathop{\rm dim} x $, then $ Sq ^ {i} x = 0 $;

5) (Cartan's formula) $ Sq ^ {i} ( xy) = \sum _ {j=} 0 ^ {i} ( Sq ^ {i} x) \cdot ( Sq ^ {i-} 1 y) $;

6) (Adem relation) if $ a < 2b $, then

$$ Sq ^ {a} Sq ^ {b} = \sum _ { t= } 0 ^ { [ } a/2] \left ( \begin{array}{c} b- t- 1 \\ a- 2t \end{array} \right ) _ {2} Sq ^ {a+} b- t Sq ^ {t} , $$

where $ ( \cdot ) _ {2} $ are binomial coefficients modulo 2.

In Cartan's formula, multiplication can be considered as outer ( $ \times $- multiplication) as well as interior (cup-multiplication). This is equivalent to the statement that the mapping $ Sq: H ^ \star ( X; \mathbf Z _ {2} ) \rightarrow H ^ \star ( X; \mathbf Z _ {2} ) $, defined by the formula

$$ Sqx = x + Sq ^ {1} x + \dots + Sq ^ {n-} 1 x + x ^ {2} ,\ \ x \in H ^ {n} ( X; \mathbf Z _ {2} ), $$

is a ring homomorphism. It follows from the stability condition that the Steenrod squares $ Sq ^ {i} $ commute with suspension and transgression.

The properties 1), 3) and 4) define the operations $ Sq ^ {i} $ uniquely and can therefore be taken as defining axioms. The constructive definition of $ Sq ^ {i} $ is based on the simplicial structure in chain groups $ C _ \star ( X) $ and on the existence of a diagonal mapping $ \Delta : X \rightarrow X \times X $. Let $ W $ be the minimal acyclic free chain $ \mathbf Z _ {2} $- complex i.e. the chain complex for which

$$ W _ {i} = \mathbf Z _ {2} [ e _ {i} , Te _ {i} ] ,\ \ de _ {i} = e _ {i-} 1 + (- 1) ^ {i} Te _ {i-} 1 , $$

where $ T $ is the generator of $ \mathbf Z _ {2} $. The existence of an equivariant chain mapping

$$ \phi : W \otimes C _ \star ( X) \rightarrow C _ \star ( X) \otimes C _ \star ( X) $$

such that

$$ \phi : ( e _ {i} \otimes \sigma ) \in C _ \star ( \sigma \otimes \sigma ) \subset C _ \star ( X) \otimes C _ \star ( X) = C _ \star ( X \times X) $$

for any simplex $ \sigma \in C _ \star ( X) $, is proved by the method of acyclic carriers or by an explicit construction (see [4]). The symbol $ C _ \star ( \sigma \otimes \sigma ) $ here signifies the smallest subcomplex of the chain complex $ C _ \star ( X) \otimes C _ \star ( X) $ containing the element $ \sigma \otimes \sigma $. Let $ i \geq 0 $. Any two cochains $ u \in C ^ {p} ( X) $, $ v \in C ^ {q} ( X) $ are put in correspondence by the formula $ ( u \cup _ {i} v)( \sigma ) = ( u \otimes v)( \phi ( e _ {i} \otimes \sigma )) $, for any simplex $ \sigma \in C _ {p+} q- i ( X) $, with the cochain $ u \cup _ {i} v \in C ^ {p+} q- i ( X) $, which is called their cup- $ i $- product. For the coboundary of this chain, the formula

$$ \delta \left ( u \cup _ { i } v \right ) = \ (- 1) ^ {i} \delta u \cup _ { i } v + (- 1) ^ {i+} p u \cup _ { i } \delta v + $$

$$ + (- 1) ^ {i+} 1 u \cup _ { i- } 1 v + (- 1) ^ {pq+} 1 v \cup _ { i- } 1 u $$

holds, from which it follows that the formula $ Sq ^ {p-} i \{ u \} = \{ u \cup _ {i} u \} $ correctly defines a homomorphism

$$ Sq ^ {p-} i : H ^ {p} ( X; \mathbf Z _ {2} ) \rightarrow H ^ {2p-} i ( X; \mathbf Z _ {2} ) $$

which does not depend on the choice of the mapping $ \phi $.

The operations $ Sq ^ {i} $ are constructed in the same way in other simplicial structures with a diagonal mapping, for example, in cohomology groups of simplicial Abelian groups, of simplicial Lie algebras, etc. However, not all properties of the Steenrod squares $ Sq ^ {i} $ are preserved then (for example, generally speaking, $ Sq ^ {0} \neq \mathop{\rm id} $) and there is yet (1991) no single general theory for the generalized operations $ Sq ^ {i} $( see [5], [6]).

Many cohomology operations which act on cohomology groups with coefficients in the groups $ \mathbf Z _ {2} $ and $ \mathbf Z _ {p} $ can be expressed in terms of the Steenrod squares and their analogues (see Steenrod reduced power). This underlines the fundamental role played by Steenrod squares in algebraic topology and its applications. For example, bordism groups are calculated using Steenrod squares.

Steenrod squares were introduced by N. Steenrod [4].

References

[1] N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[2] D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[3] M.K. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[4] N.E. Steenrod, "Products of cocylces and extensions of mappings" Ann. of Math. , 48 (1947) pp. 290–320
[5] D. Epstein, "Steenrod operations in homological algebra" Invent. Math. , 1 : 2 (1966) pp. 152–208
[6] J. May, "A general algebraic approach to Steenrod operations" , The Steenrod Algebra and Its Applications , Lect. notes in math. , 168 , Springer (1970) pp. 153–231
[7] Matematika , 5 : 2 (1961) pp. 3–11; 11–30; 30–49; 50–102

Comments

References

[a1] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
[a2] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 18
[a3] J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12
[a4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. V, Sect. 9
How to Cite This Entry:
Steenrod square. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_square&oldid=48827
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article