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The Homology of Homotopy Inverse Limits

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The Homology of Homotopy Inverse Limits by Paul G Go erss Abstract The homology of a homotopy inverse limit can b e studied by a sp ectral sequence with has E term the derived functors of limit in the category of coalgebras These derived functors can b e computed using the theory of Dieudonnemo dules if one has a diagram of connected ab elian Hopf algebras One of the standard problems in homotopy theory is to calculate the homology of a given type of inverse limit For example one might want to know the homology of the inverse limit of a tower of brations or of the pullback of a bration or of the homotopy xed p oint set of a group action or even of an innite pro duct of spaces This pap er presents a systematic metho d for dealing with this problem and works out a series of examples It simplies the foundational questions present when dealing with inverse limits to work with simplicial sets rather than top ological spaces So this pap er is written simpli cially that is a space is a simplicial set As usual this do esnt aect the homotopy theory Homology is with F co ecients p Here are some simple examples of the type of result we obtain An ab elian Hopf algebra is one for which b oth diagonal and multiplication are commutative Theorem A Let fX g b e an arbitrary set of connected nilp otent spaces and supp ose for all H X is an ab elian Hopf algebra Then there is a natural isomorphism Y Y X H H X This is a companion to a result of Bouselds where slightly dierent hypotheses Q H X is in the category of graded connected yield a slightly dierent result The pro duct coalgebras If the set is nite it is simply the graded tensor pro duct If the set is innite something more sophisticated is required See section for limits of coalgebras The author was partially supp orted by the National Science Foundation Theorem B Consider a tower of brations over the natural numbers X X X so that for all i X is connected and H X H X is a morphism of ab elian Hopf i i i algebras Supp ose each X is simple and lim X Then under either of the following i i conditions there is an isomorphism lim H X H lim X i i i for all i H X is of nite type i the tower H X H X is MittagLeer These are two of the usual conditions for vanishing of lim for ab elian groups Again lim H X must b e calculated in the category of coalgebras Under various conditions i i however this will b e isomorphic to the inverse limit as vector spaces One common example of this is if the tower fH X g is proisomorphic as a tower of Hopf algebras to a constant i i tower This is a stronger condition than MittagLeer Behind these results is a technique which I will now explain In general given a small category I and an I diagram X of spaces there is a map of graded coalgebras H holim X lim H X I I where holim is the BouseldKan homotopy inverse limit and the limit on the right is in the category of coalgebras There is no reason to think that it is either injective or surjective however it is the edge homomorphism of a sp ectral sequence This sp ectral sequence is a variant of one due to Anderson and studied by Bouseld in The ma jor advantage of our variant is that the E term can b e identied and computed by nonab elian homological algebra I the category Let CA b e the category of graded connected coalgebras over F and CA p of I diagrams in CA The p oint of this pap er is that the functor I lim CA C A I s lim and the bigraded vector has right derived functors R lim s so that R lim CA CA I I I space R lim C is a bigraded co commutative coalgebra Then we have CA I Theorem C Let X I S b e an I diagram of connected spaces Then there is a natural second quadrant homology sp ectral sequence R lim H X H holim X p CA I I Here X I S is the I diagram obtained by applying the BouseldKan pcomple p tion functor to X Because this is a second quadrant sp ectral sequence convergence is a problem For this we app eal to or This accounts for some of the hypotheses of Theorems A and B Beyond this there is the problem of computing R lim H X For CA I s example Theorems A and B are based on the assertion that R lim H X for s CA I under the listed hypotheses It is here that we use the assumption that H X I C A is actually a diagram of ab elian Hopf algebras The category of ab elian Hopf algebras is equivalent to a category of mo dules called Dieudonnemo dules and one can pass from the lim at least in some cases We give ordinary derived functors of lim for mo dules to R CA I I s s lim may not examples at least when lim for s as mo dules Even in this case R CA I I b e zero for any s Using these computations we make some calculations of the sp ectral sequence in cases where R lim H X is not concentrated in degree zero Included is an CA I example of homotopy xed p oints Finally there is the problem of relating H holim X to H holim X Again there are techniques in that apply p Despite the title of this pap er and the general thrust of this introduction this is pri marily a work in nonab elian homological algebra ve of the six sections are devoted to lim Only the sixth and last section contains homo the denitions and calculations of R CA I topy theoretic applications Sections and are devoted to the denition and homotopical algebra foundations of R lim these derived functors are in fact the cohomotopy groups CA I of a homotopy inverse limit of cosimplicial coalgebras although this language is avoided until section Section is on Dieudonnemo dules Section gives some calculations and Section contains the pro ofs of some technical lemmas The pap er b egan as a meditation on Example of spurred on by a conversation with John Hunton In fact the pro of of Theorem A given here may b e regarded as a wildly expanded version of Bouselds argument and in general this pap er owes a great debt to Finally a conversation with Bro oke Shipley on bicosimplicial spaces was useful for widening my scop e Several results for make this pap er go much more smo othly x Derived functors of limits in the category of coalgebras Let CA b e the category of graded co commutative coalgebras over a eld k Later we will sp ecialize to the case where k F As always in homotopy theory applications p commutativity is with a sign Thus if C C A and x C has diagonal x y z i i then jy j jz j i i y z z y i i i i where jw j is the degree of w Let CA CA denote the full subcategory of connected coalgebras Thus C C A if and only if C F p A useful fact is Lemma Let C CA Then C is isomorphic to the ltered colimit of its nite dimensional subcoalgebras Proof This is clear in C C A The general case follows from p I Now let I b e a small category and CA the category of I diagrams in CA By denition I lim CA C A I is right adjoint to the constant diagram functor Such limits exist for all I To see this rst supp ose I is ltered Then if C I CA is an I diagram we can form the vector k space limit lim C This is not in general a coalgebra however it is a complete coalgebra I in the following sense For i I let k E i kerflim C C ig I k Then lim C is a complete top ological vector space with resp ect to the neighborho o d base I for zero fE ig Then there is a copro duct k k k b lim C lim C lim C I I I b where denotes the completion of the tensor pro duct with resp ect to fE i E j g This is b ecause k k k b lim C lim C lim C C I I I k A subcoalgebra D lim C is a subvector space equipp ed with a copro duct D D D I making D a coalgebra and such that the evident diagram commutes w D D D u u k k k w lim C lim C lim C Then for the limit in the category of coalgebras lim C colim D I k where D runs over all subcoalgebras over lim C Because of Lemma we could equally I require the D to b e nite Next supp ose I is discrete so that lim In the ob ject I I set of I is nite then C C I I Then in general C C lim J J I where lim is in the category CA and J runs over the ltered diagram of nite subcategories of I Proposition The category CA has all limits Proof We now have pro ducts by so to get all limits we need only supply pull backs However the pullback of a diagram C C C is the cotensor pro duct C C C 12 Because of the colimit in formula lim will not in general preserve surjections I even in the case of innite pro ducts Thus one will have derived functors There are several p ossible denitions of these After some preliminaries we will give a denition in which while complicated arises in homotopy theory applications Let J CA n k b e the coaugmentation coideal functor from connected coal gebras to p ositively graded vector spaces Thus JC is nothing more than the elements of L n S W where S W k p ositive degree The functor J has a right adjoint S with SW n and n n S W W W z n where acts by p ermuting factors up to signs required by the grading If W is of nite n type SW is the free graded commutative algebra on W and hence if chark is a tensor pro duct of p olynomial and exterior algebras Let S S J CA C A b e the comp osite functor It is the underlying functor of a triple on CA Thus if C C A one has a canonical cosimplicial resolution C S C C with isomorphism induced by This can b e seen by applying J In particular S C to and noting that the resulting cosimplicial vector space has a natural contraction This resolution can b e used to dene derived functors For example if P is the primitive element functor s s R PC P S C These derived
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