View source for Lannes-T-functor

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]]).

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" />.

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.

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.

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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raises the hope that in very favourable cases the mapping

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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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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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In particular,

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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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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" />.

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003069.png" /> 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" />.

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.

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" />.

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,

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" />.

2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120030/l12003082.png" />, then

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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.

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" />.

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" />.

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

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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====
<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>