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The calculation of the homotopy type of the space of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200302.png" /> is a fundamental problem of [[Homotopy|homotopy]] theory. The set of path components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200303.png" /> corresponds to the homotopy classes of such mappings. There are relatively few cases for which this information is explicitly known (as of 1998). A major impact of the work [[#References|[a1]]] of J. Lannes on unstable modules and the T-functor has been to expand this knowledge to include many cases in which the sources and targets are classifying spaces of finite and compact Lie groups (cf. also [[Lie group|Lie group]]).
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The work of N. Steenrod and others assigns in a natural way to each [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200304.png" /> and each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200305.png" /> an algebraic model, consisting of a [[Graded algebra|graded algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200306.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200307.png" /> and an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200308.png" /> of natural operations, called the Steenrod algebra. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l1200309.png" /> induces an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003010.png" /> that commutes with the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003011.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003012.png" /> is a connected graded [[Hopf algebra|Hopf algebra]] acting on the graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003013.png" />.
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The hypothesis that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003014.png" /> is the [[Cohomology|cohomology]] of a space imposes an additional  "unstable"  condition. This is most simply stated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003015.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003016.png" /> is generated as an (non-commutative) algebra by the Steenrod operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003017.png" />, with relations forced by its actions of the cohomology of all topological spaces. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003019.png" />, the modulo-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003020.png" /> Bockstein operator. The unstable condition is then that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003023.png" />. The algebraic category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003024.png" /> of unstable algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003026.png" /> is thus an approximation to the homotopy category of topological spaces. The larger category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003027.png" /> of unstable modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003028.png" /> has also proved useful.
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The calculation of the homotopy type of the space of continuous mappings $\operatorname{Map}( X , Y )$ is a fundamental problem of [[Homotopy|homotopy]] theory. The set of path components, $\pi_0 \; \operatorname { Map } ( X , Y ) = [ X , Y ]$ corresponds to the homotopy classes of such mappings. There are relatively few cases for which this information is explicitly known (as of 1998). A major impact of the work [[#References|[a1]]] of J. Lannes on unstable modules and the T-functor has been to expand this knowledge to include many cases in which the sources and targets are classifying spaces of finite and compact Lie groups (cf. also [[Lie group|Lie group]]).
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003029.png" />, the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003030.png" /> and unstable actions are similar, but slightly more involved. However, in all cases, the set of relations in the Steenrod algebra and the unstable condition are derivable from the known action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003031.png" /> on the cohomology of products of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003032.png" />. In the following, explicit references to the coefficients are omitted.
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The work of N. Steenrod and others assigns in a natural way to each [[Topological space|topological space]] $X$ and each prime number $p$ an algebraic model, consisting of a [[Graded algebra|graded algebra]] $H ^ { * } ( X , \mathbf{F} _ { p } ) = R ^ { * }$ over $\mathbf{F} _ { p }$ and an algebra $\mathcal{A} _ { p }$ of natural operations, called the Steenrod algebra. Each $f : X \rightarrow Y$ induces an element $f ^ { * } \in \text { Hom}_{\text{alg} } ( H ^ { * } ( Y , {\bf F} _ { p } ) , H ^ { * } ( X , {\bf F} _ { p } ) )$ that commutes with the action of $\mathcal{A} _ { p }$. $\mathcal{A} _ { p }$ is a connected graded [[Hopf algebra|Hopf algebra]] acting on the graded algebra $R ^ { * }$.
  
The relationship of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003033.png" /> to its model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003034.png" /> is of particular interest. The equivalence
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The hypothesis that $R ^ { * }$ is the [[Cohomology|cohomology]] of a space imposes an additional  "unstable" condition. This is most simply stated if $p = 2$: ${\cal A} _ { 2 }$ is generated as an (non-commutative) algebra by the Steenrod operations $\{ \mathcal{S}  \operatorname {q}  ^ { i } : i \geq 0 \}$, with relations forced by its actions of the cohomology of all topological spaces. For example, $\mathcal{S} \text{q} ^ { 0 } = \operatorname{Id}$ and $\mathcal{S} \text{q} ^ { 1 } = \beta$, the modulo-$2$ Bockstein operator. The unstable condition is then that ${\cal S} \operatorname {q} ^ { i } x _ { n } = 0$ for $i &gt; n$ and $\mathcal{S} \operatorname{q} ^ { n } x _ { n } = x _ { n } ^ { 2 }$. The algebraic category $\mathcal{K}$ of unstable algebras $\{ \mathcal{R} ^ { * } \}$ over $\mathcal{A} _ { p }$ is thus an approximation to the homotopy category of topological spaces. The larger category $\mathcal U$ of unstable modules over $\mathcal{A} _ { p }$ has also proved useful.
  
<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/l/l120/l120030/l12003035.png" /></td> </tr></table>
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For $p &gt; 2$, the structure of $\mathcal{A} _ { p }$ and unstable actions are similar, but slightly more involved. However, in all cases, the set of relations in the Steenrod algebra and the unstable condition are derivable from the known action of $\mathcal{A} _ { p }$ on the cohomology of products of copies of $B {\bf Z} / p {\bf Z}$. In the following, explicit references to the coefficients are omitted.
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The relationship of $\pi _0 \operatorname { Map } ( X , Y )$ to its model $\operatorname{Hom}_{ \mathcal{K} } ( H ^ { * } ( Y , \mathbf{F} _ { p } ) , H ^ { * } ( X , \mathbf{F} _ { p } ) )$ is of particular interest. The equivalence
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\begin{equation*} \operatorname { Map } ( X \times Z , Y ) \rightarrow \operatorname { Map } ( X , \operatorname { Map } ( Z , Y ) ) \end{equation*}
  
 
raises the hope that in very favourable cases the mapping
 
raises the hope that in very favourable cases the mapping
  
<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/l/l120/l120030/l12003036.png" /></td> </tr></table>
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\begin{equation*} \operatorname{Hom}_{\mathcal{K}} ( H ^ { * } \operatorname { Map } ( Z , Y ) , H ^ { * } X ) \rightarrow \end{equation*}
  
<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/l/l120/l120030/l12003037.png" /></td> </tr></table>
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\begin{equation*}  \rightarrow \operatorname{Hom}_{\mathcal{K}} ( H ^ { * } Y , H ^ { * } X \bigotimes H ^ { * } Z ) \end{equation*}
  
might be an isomorphism. That suggests that in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003039.png" /> should be approximated by the left adjoint functor to tensoring on the right by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003040.png" />. This motivated J. Lannes to define the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003042.png" /> as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003043.png" /> is a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003044.png" />-vector space, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003045.png" />-functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003046.png" /> is the left adjoint in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003047.png" /> of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003048.png" />. In the topological case, there is a natural mapping
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might be an isomorphism. That suggests that in the category $\mathcal{K}$, $H ^ { * } \operatorname { Map } ( Z , Y )$ should be approximated by the left adjoint functor to tensoring on the right by $H ^ { * } Z$. This motivated J. Lannes to define the functor $T$ as follows: If $E$ is a finite-dimensional $\mathbf{F} _ { p }$-vector space, then the $T$-functor $T _ { E } : \mathcal{U} \rightarrow \mathcal{U} $ is the left adjoint in $\mathcal U$ of the functor $( ( _- ) \otimes _ {{\bf F}_p }  H ^ { * } B V ) :\cal U \rightarrow U$. In the topological case, there is a natural mapping
  
<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/l/l120/l120030/l12003049.png" /></td> </tr></table>
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\begin{equation*} \lambda _ { X } : T _ { E } H ^ { * } X \rightarrow H ^ { * } \operatorname { Map } ( B E , X ). \end{equation*}
  
For general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003050.png" />, the adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003051.png" /> accounts for only part of the starting page of a Bousfield–Kan unstable Adams spectral sequence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003052.png" />. Lannes provides the basic connection to topology by blending the algebraic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003054.png" /> with the Bousfield–Kan spectral sequence: For many interesting spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003055.png" />,
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For general $Z$, the adjoint to $( ( \_ ) \otimes _ { \mathbf{F}_p }  H ^ { * } Z )$ accounts for only part of the starting page of a Bousfield–Kan unstable Adams spectral sequence for $\operatorname{Map}( Z , Y )$. Lannes provides the basic connection to topology by blending the algebraic properties of $T _ { E }$ and $\mathcal{K}$ with the Bousfield–Kan spectral sequence: For many interesting spaces $X$,
  
<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/l/l120/l120030/l12003056.png" /></td> </tr></table>
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\begin{equation*} H ^ { * } \operatorname { Map } ( B E , X ) \approx T _ { E } H ^ { * } X. \end{equation*}
  
 
In particular,
 
In particular,
  
<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/l/l120/l120030/l12003057.png" /></td> </tr></table>
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\begin{equation*} \pi _ { 0 } \operatorname { Map } ( B E , X ) = [ B E , X ] = \operatorname { Hom } _ { \mathcal{K} } ( H ^ { * } X , H ^ { * } B E ). \end{equation*}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003058.png" />, one has the path component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003059.png" /> of functions homotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003060.png" />. The analogous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003061.png" />-construct is as follows: Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003062.png" /> induces a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003063.png" />-module structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003064.png" /> and
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For $f : X \rightarrow Y$, one has the path component $\operatorname{Map}( X , Y ) _ { f }$ of functions homotopic to $f$. The analogous $T$-construct is as follows: Each $\varphi \in \operatorname{Hom}_{\mathcal{K}}( R ^ { * } , H ^ { * } B E )$ induces a $T ^ { 0 } E$-module structure on $\mathbf{F} _ { p }$ 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/l/l120/l120030/l12003065.png" /></td> </tr></table>
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\begin{equation*} T _ { E , \varphi } R ^ { * } = T _ { E } R ^ { * } \bigotimes _ { T ^ { 0 } E } \mathbf{F} _ { p }. \end{equation*}
  
The most striking features of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003066.png" /> are summarized below (see also [[#References|[a1]]]). To some extent, these were presaged by work of G. Carlsson and H.T. Miller, who established that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003067.png" /> are injectives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003068.png" />.
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The most striking features of $T _ { E }$ are summarized below (see also [[#References|[a1]]]). To some extent, these were presaged by work of G. Carlsson and H.T. Miller, who established that the $\{ H ^ { * } B V \}$ are injectives in $\mathcal U$.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003069.png" /> is exact.
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a) $T _ { E }$ is exact.
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003070.png" /> respects tensor products, i.e <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003071.png" />.
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b) $T _ { E }$ respects tensor products, i.e $T _ { E } ( M \otimes _ { \mathbf{F}_ p} N) = T _ { E } M \otimes _ { \mathbf{F}_ p}  T _ { E } N$.
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003072.png" /> commutes with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003073.png" />th power operations in a suitable sense.
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c) $T _ { E }$ commutes with the $p$th power operations in a suitable sense.
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003074.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003075.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003076.png" />.
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d) $T _ { E }$ maps $\mathcal{K}$ to $\mathcal{K}$.
  
In principle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003077.png" /> can be calculated by using the exactness property and a resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003078.png" /> by free unstable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003079.png" />-modules. In practice, other methods are often more effective; for example,
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In principle, $T _ { E } M ^ { * }$ can be calculated by using the exactness property and a resolution of $M ^ { * }$ by free unstable $\mathcal{A} _ { p }$-modules. In practice, other methods are often more effective; for example,
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003080.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003081.png" />.
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1) If $M ^ { * }$ is finite, then $T _ { E } M ^ { * } = M ^ { * }$.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003082.png" />, then
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2) If $R ^ { * } = H ^ { * } B V$, then
  
<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/l/l120/l120030/l12003083.png" /></td> </tr></table>
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\begin{equation*} T _ { E } R ^ { * } = \prod _ { \text { Hom}_{ \text{grp} } ( E , V ) } H ^ { * } B V, \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003085.png" /> finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003086.png" />-vector spaces.
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for $E$ and $V$ finite-dimensional $\mathbf{F} _ { p }$-vector spaces.
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003087.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003088.png" /> is an inclusion, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003089.png" /> is the smallest sub-Hopf algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003090.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003091.png" />.
+
3) If $\tau : R ^ { * } \rightarrow H ^ { * } B E$ in $\mathcal{K}$ is an inclusion, then $T _ { E , \tau } R ^ { * }$ is the smallest sub-Hopf algebra of $H ^ { * } B E$ that contains $\tau ( R ^ { * } )$.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003092.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003093.png" />-complex with fixed point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003095.png" /> is the modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003096.png" /> cohomology of the Borel construction, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003097.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003098.png" />.
+
4) If $X$ is a finite $E$-complex with fixed point set $X ^ { E }$ and $H ^ { *_{E}} X$ is the modulo $p$ cohomology of the Borel construction, then $T_{E, \text{id}} H _ { E } ^ { * } X = H ^ { * } B E \otimes _ { \text{F}_ p } H ^ { * } X ^ { E }$ in $\mathcal{K}$.
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003099.png" /> is a compact [[Lie group|Lie group]], then
+
5) If $G$ is a compact [[Lie group|Lie group]], then
  
<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/l/l120/l120030/l120030100.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l120030100.png"/></td> </tr></table>
  
 
These examples each have powerful topological consequences. For example, the first and fourth lead to new proofs of the Sullivan conjecture, originally proved by Miller and Carlsson. The last leads to a new view of the homotopy theory of classifying spaces. Most of the above is referenced in [[#References|[a2]]].
 
These examples each have powerful topological consequences. For example, the first and fourth lead to new proofs of the Sullivan conjecture, originally proved by Miller and Carlsson. The last leads to a new view of the homotopy theory of classifying spaces. Most of the above is referenced in [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Lannes,  "Sur les espaces fonctionnels dont la source est le classifiant d'un <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l120030101.png" />-groupe abélien élémentaire"  ''Inst. Hautes Etudes Sci. Publ. Math.'' , '''75'''  (1992)  pp. 135–244  (Appendix by M. Zisman)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schwartz,  "Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture" , Univ. Chicago Press  (1994)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  J. Lannes,  "Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire"  ''Inst. Hautes Etudes Sci. Publ. Math.'' , '''75'''  (1992)  pp. 135–244  (Appendix by M. Zisman)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L. Schwartz,  "Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture" , Univ. Chicago Press  (1994)</td></tr></table>

Revision as of 17:00, 1 July 2020

The calculation of the homotopy type of the space of continuous mappings $\operatorname{Map}( X , Y )$ is a fundamental problem of homotopy theory. The set of path components, $\pi_0 \; \operatorname { Map } ( X , Y ) = [ X , Y ]$ corresponds to the homotopy classes of such mappings. There are relatively few cases for which this information is explicitly known (as of 1998). A major impact of the work [a1] of J. Lannes on unstable modules and the T-functor has been to expand this knowledge to include many cases in which the sources and targets are classifying spaces of finite and compact Lie groups (cf. also Lie group).

The work of N. Steenrod and others assigns in a natural way to each topological space $X$ and each prime number $p$ an algebraic model, consisting of a graded algebra $H ^ { * } ( X , \mathbf{F} _ { p } ) = R ^ { * }$ over $\mathbf{F} _ { p }$ and an algebra $\mathcal{A} _ { p }$ of natural operations, called the Steenrod algebra. Each $f : X \rightarrow Y$ induces an element $f ^ { * } \in \text { Hom}_{\text{alg} } ( H ^ { * } ( Y , {\bf F} _ { p } ) , H ^ { * } ( X , {\bf F} _ { p } ) )$ that commutes with the action of $\mathcal{A} _ { p }$. $\mathcal{A} _ { p }$ is a connected graded Hopf algebra acting on the graded algebra $R ^ { * }$.

The hypothesis that $R ^ { * }$ is the cohomology of a space imposes an additional "unstable" condition. This is most simply stated if $p = 2$: ${\cal A} _ { 2 }$ is generated as an (non-commutative) algebra by the Steenrod operations $\{ \mathcal{S} \operatorname {q} ^ { i } : i \geq 0 \}$, with relations forced by its actions of the cohomology of all topological spaces. For example, $\mathcal{S} \text{q} ^ { 0 } = \operatorname{Id}$ and $\mathcal{S} \text{q} ^ { 1 } = \beta$, the modulo-$2$ Bockstein operator. The unstable condition is then that ${\cal S} \operatorname {q} ^ { i } x _ { n } = 0$ for $i > n$ and $\mathcal{S} \operatorname{q} ^ { n } x _ { n } = x _ { n } ^ { 2 }$. The algebraic category $\mathcal{K}$ of unstable algebras $\{ \mathcal{R} ^ { * } \}$ over $\mathcal{A} _ { p }$ is thus an approximation to the homotopy category of topological spaces. The larger category $\mathcal U$ of unstable modules over $\mathcal{A} _ { p }$ has also proved useful.

For $p > 2$, the structure of $\mathcal{A} _ { p }$ and unstable actions are similar, but slightly more involved. However, in all cases, the set of relations in the Steenrod algebra and the unstable condition are derivable from the known action of $\mathcal{A} _ { p }$ on the cohomology of products of copies of $B {\bf Z} / p {\bf Z}$. In the following, explicit references to the coefficients are omitted.

The relationship of $\pi _0 \operatorname { Map } ( X , Y )$ to its model $\operatorname{Hom}_{ \mathcal{K} } ( H ^ { * } ( Y , \mathbf{F} _ { p } ) , H ^ { * } ( X , \mathbf{F} _ { p } ) )$ is of particular interest. The equivalence

\begin{equation*} \operatorname { Map } ( X \times Z , Y ) \rightarrow \operatorname { Map } ( X , \operatorname { Map } ( Z , Y ) ) \end{equation*}

raises the hope that in very favourable cases the mapping

\begin{equation*} \operatorname{Hom}_{\mathcal{K}} ( H ^ { * } \operatorname { Map } ( Z , Y ) , H ^ { * } X ) \rightarrow \end{equation*}

\begin{equation*} \rightarrow \operatorname{Hom}_{\mathcal{K}} ( H ^ { * } Y , H ^ { * } X \bigotimes H ^ { * } Z ) \end{equation*}

might be an isomorphism. That suggests that in the category $\mathcal{K}$, $H ^ { * } \operatorname { Map } ( Z , Y )$ should be approximated by the left adjoint functor to tensoring on the right by $H ^ { * } Z$. This motivated J. Lannes to define the functor $T$ as follows: If $E$ is a finite-dimensional $\mathbf{F} _ { p }$-vector space, then the $T$-functor $T _ { E } : \mathcal{U} \rightarrow \mathcal{U} $ is the left adjoint in $\mathcal U$ of the functor $( ( _- ) \otimes _ {{\bf F}_p } H ^ { * } B V ) :\cal U \rightarrow U$. In the topological case, there is a natural mapping

\begin{equation*} \lambda _ { X } : T _ { E } H ^ { * } X \rightarrow H ^ { * } \operatorname { Map } ( B E , X ). \end{equation*}

For general $Z$, the adjoint to $( ( \_ ) \otimes _ { \mathbf{F}_p } H ^ { * } Z )$ accounts for only part of the starting page of a Bousfield–Kan unstable Adams spectral sequence for $\operatorname{Map}( Z , Y )$. Lannes provides the basic connection to topology by blending the algebraic properties of $T _ { E }$ and $\mathcal{K}$ with the Bousfield–Kan spectral sequence: For many interesting spaces $X$,

\begin{equation*} H ^ { * } \operatorname { Map } ( B E , X ) \approx T _ { E } H ^ { * } X. \end{equation*}

In particular,

\begin{equation*} \pi _ { 0 } \operatorname { Map } ( B E , X ) = [ B E , X ] = \operatorname { Hom } _ { \mathcal{K} } ( H ^ { * } X , H ^ { * } B E ). \end{equation*}

For $f : X \rightarrow Y$, one has the path component $\operatorname{Map}( X , Y ) _ { f }$ of functions homotopic to $f$. The analogous $T$-construct is as follows: Each $\varphi \in \operatorname{Hom}_{\mathcal{K}}( R ^ { * } , H ^ { * } B E )$ induces a $T ^ { 0 } E$-module structure on $\mathbf{F} _ { p }$ and

\begin{equation*} T _ { E , \varphi } R ^ { * } = T _ { E } R ^ { * } \bigotimes _ { T ^ { 0 } E } \mathbf{F} _ { p }. \end{equation*}

The most striking features of $T _ { E }$ are summarized below (see also [a1]). To some extent, these were presaged by work of G. Carlsson and H.T. Miller, who established that the $\{ H ^ { * } B V \}$ are injectives in $\mathcal U$.

a) $T _ { E }$ is exact.

b) $T _ { E }$ respects tensor products, i.e $T _ { E } ( M \otimes _ { \mathbf{F}_ p} N) = T _ { E } M \otimes _ { \mathbf{F}_ p} T _ { E } N$.

c) $T _ { E }$ commutes with the $p$th power operations in a suitable sense.

d) $T _ { E }$ maps $\mathcal{K}$ to $\mathcal{K}$.

In principle, $T _ { E } M ^ { * }$ can be calculated by using the exactness property and a resolution of $M ^ { * }$ by free unstable $\mathcal{A} _ { p }$-modules. In practice, other methods are often more effective; for example,

1) If $M ^ { * }$ is finite, then $T _ { E } M ^ { * } = M ^ { * }$.

2) If $R ^ { * } = H ^ { * } B V$, then

\begin{equation*} T _ { E } R ^ { * } = \prod _ { \text { Hom}_{ \text{grp} } ( E , V ) } H ^ { * } B V, \end{equation*}

for $E$ and $V$ finite-dimensional $\mathbf{F} _ { p }$-vector spaces.

3) If $\tau : R ^ { * } \rightarrow H ^ { * } B E$ in $\mathcal{K}$ is an inclusion, then $T _ { E , \tau } R ^ { * }$ is the smallest sub-Hopf algebra of $H ^ { * } B E$ that contains $\tau ( R ^ { * } )$.

4) If $X$ is a finite $E$-complex with fixed point set $X ^ { E }$ and $H ^ { *_{E}} X$ is the modulo $p$ cohomology of the Borel construction, then $T_{E, \text{id}} H _ { E } ^ { * } X = H ^ { * } B E \otimes _ { \text{F}_ p } H ^ { * } X ^ { E }$ in $\mathcal{K}$.

5) If $G$ is a compact Lie group, then

These examples each have powerful topological consequences. For example, the first and fourth lead to new proofs of the Sullivan conjecture, originally proved by Miller and Carlsson. The last leads to a new view of the homotopy theory of classifying spaces. Most of the above is referenced in [a2].

References

[a1] J. Lannes, "Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire" Inst. Hautes Etudes Sci. Publ. Math. , 75 (1992) pp. 135–244 (Appendix by M. Zisman)
[a2] L. Schwartz, "Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture" , Univ. Chicago Press (1994)
How to Cite This Entry:
Lannes-T-functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lannes-T-functor&oldid=50376
This article was adapted from an original article by Clarence W. Wilkerson, Jr. (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article