© 1999 Birkhauser Verlag, Basel Comment. Math. Helv. 74 (1999) 657-670 0010-2571/99/040657-13 $ 1.50+0.20/0 I Commentarii Mat hematic! Helvetic! The flniteness obstruction for loop spaces D. Notbohm Abstract. For finitely dominated spaces, Wall constructed a finiteness obstruction, which de cides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite //-spaces, that are //-spaces whose homol- ogy looks like the homology of a finite CW-complex. In this paper we prove this conjecture for loop spaces. In particular, this shows that every quasi finite loop space is actually homotopy equivalent to a finite CW-complex. Mathematics Subject Classification (1991). 57Q12, 55R35, 55R10 Keywords. Finiteness obstruction, Wall obstruction, loop space, p-compact group 1. Introduction A topological space is called quasi-finite, if the direct sum QnHn(X',%) of all integral homology groups is a finitely generated abelian group. Then you can ask whether the space X is (weakly) homotopy equivalent to a finite CW-complex. For finitely dominated spaces, Wall constructed an invariant which decides this problem [29] [30]. A space X is finitely dominated, if there exist a finite CW- complex K and maps / : K — > X and g : X — > K such that the composition fg is homotopic to the identity idx on X. One can show that finitely dominated spaces are always homotopy equivalent to CW-complexes and that each component is path connected (e.g. see [17, Section 2]). Examples of finitely dominated spaces are given by quasi-finite nilpotent spaces. Actually, for nilpotent space, both conditions, finitely dominated and quasi-finite are equivalent [15]. A space X is called nilpotent if X is homotopy equivalent to a CW-complex and if 7ti(X) acts nilpotent on the homotopy groups 7r*(X). In particular, 7Ti(X) is a nilpotent group. The space X is called simple if, in addition, the action is trivial. In this case, Ki(X) is abelian. For a finitely dominated space X, the finiteness obstruction &(X) of Wall is an element in the reduced projective class group K$(L[KI(X)}) of the integral group ring. And &(X) vanishes if and only if X is homotopy equivalent to a finite CW-complex [29] [30]. It is interesting to investigate the question how topological  658 D. Notbohm CMH properties of spaces are reflected by properties of the finiteness obstruction. For instance, it is not known, but conjectured, that the finiteness obstruction for quasi- finite loop spaces or for ff-spaces always vanishes. In this paper we will prove this conjecture for loop spaces. A loop space L consists of a triple (I/, >!/, e), where L and BL are of the homotopy type of a CW-complex, BL pointed, and where e : QBL — > L is a homotopy equivalence between the loop space of BL and L. Examples of loop spaces are given by topological groups and their classifying spaces. Loop spaces inherit properties from the space I/; e.g. L is called quasi-finite, if the space L is so. Loop spaces are always nilpotent spaces. In fact, they are simple. Thus every quasi-finite loop space is finitely dominated and the finiteness obstruction &(L) is defined. Theorem 1.1. Let L be a quasi finite loop space. Then 0 = &(L) G and L is homotopy equivalent to a finite CW -complex. For a quasi-finite nilpotent space X, Mislin proved results of this type if is infinite [16] or if 7ri(X) is cyclic of prime power order [15]. The proof of our result is a consequence of the next two statements. For every fibration F — > E — » >, the fiber transport along loops in B establishes a homomorphism 7Ti(B) — » [F, F] from the fundamental group of B into the set of homotopy classes of self maps of the fiber F [31, p. 98 ff]. The fibration is called oriented if this homomorphism is trivial. Theorem 1.2. Let L be a quasi-finite loop space, such that ^i(L) is finite. Let r := dirriQH^(BL, Q). Then there exist a semi simple connected compact Lie group G and an oriented fibration G^L^E such that the following holds: (1) For the universal cover G of G, we have G = SU(2)r. (2) The space E is simple and quasi-finite. Theorem 1.3. Let X be a finitely dominated space, such that 7ti(X) is finite. If, up to homotopy, there exists an oriented fibration G ^X -> B such that G ^ {*} is a connected compact Lie group and B a finitely dominated space, then, the finiteness obstruction u(X) vanishes and X is homotopy equivalent to a finite complex. For principal G-fibrations X — > B Theorem 1.3 is proved in [22]. In the general case, the statement is known and a simple consequence of some vanishing results  Vol. 74 (1999) The finiteness obstruction for loop spaces 659 of Luck for the algebraic transfer in algebraic K-theory associated to a fibration [14] (see Section 2). We couldn't find a reference for it, but believe that it is worth to be stated. Proof of Theorem 1.1. If 7ri(L) is infinite, we can apply the above mentioned result of Mislin [16]. If 7r(L) is finite, the statement follows from Theorem 1.2 and Theorem 1.3, since every quasi finite nilpotent space is finitely dominated [16]. D Theorem 1.2 is a corollary or a weak version of the next statement. Theorem 1.4. Let L be a quasi- finite loop space, such that 7ri(L) is finite. Let r := dimqH4:(BL, Q). Then there exist a semi simple connected compact Lie group G, quasi-finite loop spaces M and N and a fibration E -> BM -> BN such that the following holds: (1) For the universal cover G of G, we have G = SU(2)r. (2) The spaces G and M are homotopy equivalent as well as L and N. (3) The space E is simple and quasi-finite. The proof of Theorem 1.4 goes as follows. Since completion turns the quasi- finite loop space L into a p-compact group (I/^, BL^, e^) we can use the theory of p-compact groups to construct the above fibration at each prime (see Section 3). Localized at 0, the existence of the fibration is basically a consequence of the fact that the rationalization BL$ is equivalent to a product of rational Eilenberg- MacLane spaces. Using the arithmetic square we glue all these data together and get the fibration claimed in Theorem 1.4. In Section 2, we prove Theorem 1.3. The other sections are devoted to the proof of Theorem 1.4. In Section 3, we prove the p-completed version of Theorem 1.4. In Section 4 we recall some material about the arithmetic square and the genus of a space. Section 5 contains the construction of the fibration of Theorem 1.4. The analysis of the genus of those compact Lie groups appearing in Theorem 1.4 is worked out in section 6 and will complete the proof of Theorem 1.4. We will switch between the p-adic completion of Bousfield and Kan [2] and the p-profinite completion of Sullivan [28]. But, for nilpotent spaces with mod-p homology of finite type, both constructions coincide [2]. It is a pleasure thank Wolfgang Luck for telling me that a result like Theorem 1.2 implies the vanishing of the finiteness obstruction for quasi finite loop spaces and for several helpful discussions on this subject. I am also grateful to the SFB 478 " Geometrische Strukturen in der Mathematik" at Minister for its hospitality when part of this work was done.  660 D. Notbohm CMH 2. Proof of Theorem 1.3 For a fibration F -^ E — » B with finitely dominated total space, base and fiber, Ehrlich [11] constructed a geometric transfer p- : KQ(Z[7n(B)] ^ KQ(Z[7n(E)] for the projective class groups of the integral group rings of the fundamental groups (see also [25]). The map p- also gives a transfer for the reduced projective class groups. Ehrlich also proved a formula relating the finiteness obstruction of E and B [12], namely where x(B) denotes the Euler characteristic of B and where i* : KQ(Z[KI(F)} — » KQ(L[KI(E)} is the map induced by induction. In [13] [14], for fibrations of the above type, Luck constructed an algebraic transfer, showed that both coincide and proved the following vanishing result. Theorem 2.1. (Luck [14, 9.1]) Let G — > E — > B be an oriented fibration, such that G is a connected compact Lie group. If 7Ti(B) is finite, then the algebraic transfer vanishes. Proof of Theorem 1.3. The compact Lie group G is connected and homotopy equivalent to a finite CW-complex. Hence o)(G) = 0. Moreover, 7ri(B) is finite since 7ti(X) is. Thus, the statement is a simple consequence of Theorem 2.1 and the above formula. D 3. Particular subgroups of p compact groups A p-compact group X is a loop space X = (X, BX, e) such that BX is p-complete and pointed and such that X is IF^-finite, i.e. H*(X;Wp) is finite in each degree and vanishes in almost all degrees. As turned out [6] [7] [20] p-compact groups behave very much like compact Lie groups. In particular, most of the classical notions for compact Lie groups are available in this context; e.g. there exist subgroups, maximal tori and Weyl groups with the same properties and we can speak of centralizers of p-compact toral subgroups. A p-compact toral group P is a p-compact group P such that TTQ(P) = TTi(BP) is a finite p-group and such that the universal cover of BP is equivalent to an Eilenberg-MacLane space of the form K(Zpn, 2). For details and further notions we refer the reader to the above mentioned references and/or to the survey articles [5], [19] and [24].  Vol. 74 (1999) The finiteness obstruction for loop spaces 661 In this section we want to prove the following statement, which is a p-completed version of Theorem 1.4. Proposition 3.1. Let X be a connected p-compact group, such that iri(X) is finite. Let r := dim^H^(BX\ Z£) Q. Then there exists a connected compact Lie group G and a map -^ BX such that the following holds: (1) For the universal cover G of G, we have G = SU(2)r. If p is odd, we can choose G = SU(2)r. (2) The induced map H4(BX; Z£) ® Q — > #4(5G£; Z£) (g) Q is an isomorphism. (3) The homotopy fiber X/Gp of f is simple and¥p-finite. Before we start with the proof, we draw one corollary. For a space Y we denote by Y (n) the n-th Postnikov section and by Y(ri) the n-connected cover of Y. Hence these spaces fit into a fibration Y(ri) -> Y -> Y(n) . Corollary 3.2. The composition BGp — » BX — > BX(4) induces an equivalence 0 -^ BX(4)Q. Proof. Since 7Ti(G) and 7ri(X) are finite and since G = SU(2)r, both spaces, (BGp)o and BX(4)o, are Eilenberg-MacLane spaces with non vanishing 4-th ho motopy group. The universal coefficient theorem shows that H^(BX\Jjp) (g) Q is the dual of #4(5X;Z£)(g)Q and hence, that H±(BG^ Z£) X of p-compact groups. This might not be a monomorphism, but the kernel K C SU(2)p of this homomorphism is a finite p-group and a central subgroup of SU(2)^ actually of SU(2) in the honest sense of compact Lie groups [23]. The kernel can be divided out [23] and, doing this, yields a monomorphism (SU(2)/K)p — » X respective ly a map B(SU(2)/K) — > BX. The last claim follows from [6]. The center Z(SU(2) ^ Z/2 is a 2-group. Thus, for odd primes, K = 0 and for p = 2 the quotient G := SU(2)/K is a connected compact Lie group isomorphic to SU(2) or 5O(3). In both cases, there exists a connected compact Lie group G and a fibration X/Gp — > BGp — > BX satisfying condition (1) of the statement. The homotopy fiber X/Gp is simple since the homotopy fiber of a map between simply connected spaces is. Moreover, X/Gp is IF^-finite since G — » X is a monomorphism [6]. This establishes part (3). Now we consider, the induced homomorphism $) (g) Q ^ . By construction, the Wx-invariant quadratic form qx is mapped onto a WG- invariant quadratic form qf G H4(BG'^ ) (g) Q. If qr = 0, the map BG — > BX induces the trivial map in reduced rational cohomology [1] and is therefore null homotopic [18]. This contradicts the fact that X/Gp is IF^-finite and finishes the proof of the statement for simple simply connected p-compact groups. Now let X be a connected p-compact group, such that 7ri(X) is finite. Then the universal cover X is again a p-compact group and establishes a map BX — » BX which induces an isomorphism H*(BX',%£) (g) Q ^ H*(BX',%£) (g) Q [20]. By [8] , X = Yli Xi splits into a finite product of simply connected simple p- compact groups Xif For each 2, we choose a connected compact Lie group GI and a monomorphism dp -^ Xi satisfying the statement. Taking the product gives a homomorphism  Vol. 74 (1999) The finiteness obstruction for loop spaces 663 which might not be a monomorphism. But, analogously as above, the kernel K is a central subgroup of G (in the Lie group sense) and, by dividing out the kernel, the homomorphism can be made into a monomorphism GA := (Gf /K)A — » X. Again, the center of G' is a 2-group. Hence, for odd primes, G = SU(2)r. Moreover, G is a connected compact Lie group satisfying part (1) of the statement. By construction, all the maps establish a commutative diagram  > BX BX where the vertical arrows induce isomorphisms in rational cohomology. The top horizontal map satisfies part (2) and so does the bottom arrow. By the same argument as above the homotopy fiber X/GA of the bottom horizontal arrow is simple and IF^-finite. D 4. The arithmetic square and genus sets Most of the material in this section is taken from [28] and [32]. For a space X we define XA := Hp-^ to be the product of the p-adic completions in the sense of Bousfield and Kan [2] taken over all primes. For nilpotent spaces with mod-p homology of finite type, this is equivalent to the profinite completion of Sullivan [28] [2]. Sullivan also constructed a formal completion for spaces [28]. For a space X, the p- formal completion Xp is defined to be the homotopy colimit hocolimiXiA where Xi ranges over all finite subcomplexes of X. The formal completion X^ is given by hocolirriiXiA. Both completions are coaugmented functors on the ho motopy category of topological spaces. That is they come with a map X — > X^ respectively X — > XA. By XQ we denote the formal completion of the rationaliza tion of X and by XQ the rationalization of the profinite completion of X. Theorem 4.1. ([Sullivan]) Let X be a connected simple CW -complex of finite type. Then (1)X*Q~X$. (2) The diagram ^0 — ^o is a fiber square up to homotopy.  664 D. Notbohm CMH Proof. Actually, in [28] this is only proved for simply connected spaces, but all proofs also work in the case of simple spaces (see also [2]). D Remark 4.2. For a simple space X, we can identify XQ and XQ via the above equivalence. The homotopy groups TT*(XQ) are modules over Q(g)ZA and therefore topological groups. For simple spaces X and y, a map / : (^A)o — > (^A)o is called TT* -continuous if TT*(/) is continuous. In the finitely generated case this is equivalent to TT*(/) being Q ® ZA-linear. If / is induced from a map XQ — > YQ by formal completion or from XA — > YA by localization then / is always TT*- continuous. If X$ ~ yQA we denote by HEC$(X,Y) the set of homotopy classes of TT*- continuous homotopy equivalences / : (^A)o — » 0^A)o- If X = Y, then we define HECA(X) := HECA(X, X). The elements of these groups are called gluing maps. For any element a G HEC$(X,Y) we define W := W(X,a,Y) to be the homotopy pull back of Proposition 4.3. Let X and y are simple spaces of finite type and let a G HECA(X, y). TAen, /or VF := VF(X, a, y), tte following holds: (1) WA -yA and WQ ~^0. (2) W is a simple space of finite type. Proof. Part (1) is obvious. By [32; 3.7], the abelian groups 7rn(W) are finitely generated for all n and, since 7rn(W) — > 7rn(WA) = 7rn(XA) is a monomorphism, W is simple. This proves the second part of the statement. D If X = Y is a nilpotent space of finite type, the above construction gives a map HECA(X) — > G$(X) where the genus G$(X) of X is the set of all homotopy equivalence classes of nilpotent spaces Y of finite type such that yA ~ XA and yo — XQ. In [32; 3.8] is proved that the above assignment identifies the genus set of X with a particular double coset of HECA(X). 5. Construction of a particular fibration Let L be a a quasi-finite connected loop space with finite fundamental group. Then, the rational cohomology H*(BL; Q) of the classifying space BL of L is a finitely generated polynomial algebra with generators #1, . . . , xn. The generators are cho sen as images of non divisible integral classes, also denoted by Xi G #*(£>!/; Z). These integral classes define a map BL A K :=  Vol. 74 (1999) The finiteness obstruction for loop spaces 665 where K(L,\Xi\) is an Eilenberg-MacLane space, where Xi\ denotes the degree of the class Xi (which in our case is always even) and where ^(4), as the 4-th Postnikov section, is the product of all factors of degree less than 5 and -K"(4), as the 4-th connected cover, the product of all other factors. Rationally, this map establishes equivalences BL$ — > KQ ~ K(4)o x K(4)o and BL^ ~ BLbQ — > KQ ~ K ~ K<4} x tf(4)£. Let r := dim^H^(BL; Q) = dim^H^(BL; Z£)®Q. Completed at every prime, I/ becomes a p-compact group. By Proposition 3.1 and Corollary 3.2, there exist a semi simple connected compact Lie group G, such that G = SU(2)r^ and, for every prime p a map BGA — > BL$ such that the composition BGA — > BLA — > BL(4)A induces an equivalence after localizing at 0. Putting all these maps together we get a map / : BGA — > BLA such that BG^ — > BL(4)^ is an equivalence. Since >Go, >I/(4)o and ^{4}o are rational Eilenberg-MacLane spaces with isomorphic homotopy groups, all these spaces are homotopy equivalent. Thus, there exists a map s : BG$ — > >I/o, given by a section of BL$ — » ^{4}o such that » K(4)o is null homotopic. We have a diagram where the vertical arrows are the obvious homotopy equivalences, where pr\ de notes the projection onto the first factor and where the composition of the hor izontal arrows is a homotopy equivalence. The compositions es =: (51,52) and ef =: (/i, /2) can be written as two components. Confusing notations, e, / and s as well as fa and s^ also denote the maps between the uncompleted or unlocalised spaces if they exist. We want to change the gluing map j : BL^ — > ^L^ in such a way that it lifts to the gluing map a := (prief)~lpries : BG^ — » BG^. This map is a homotopy equivalence and TT* -continuous since all homotopy groups of BG and BL are finitely generated and since all maps are Q (8) ZA -linear. As a product of Eilenberg-MacLane spaces, BL^ is an H-space. The product of the two maps and /32 := j\ defines a map f3 := fr • fo : BLb0  666 D. Notbohm CMH which, by construction, is TT* -continuous and fits into a homotopy commutative diagram Here I/o/Go and I/A/GA denote the homotopy fibers of the map s : BGo — > and / : >GA — » >GA. The maps a and (3 are homotopy equivalences and so is 7. Since / and 5 have left inverse, the map 7 is also TT* -continuous. The homotopy groups of I/o /Go are finite dimensional rational vector spaces and the homotopy groups of I/A/GA finitely generated ZA-modules. These considerations lead to the following proposition. Proposition 5.1. Let L be a connected quasi-finite loop space such that 7ri(L) is finite. Let r := dimQH^(BL'^Q). Then, there exist a semi simple connected compact Lie group G; quasi-finite connected loop spaces M and N and a fibration E -> BM -> BN such that the following holds: (1) E is quasi-finite and simple. (3) BM is in the genus of BG (4) BN is in the genus of BL. Proof. Using the arithmetic square, the above diagram establishes a fibration E — » BM — > BN. The last three properties are true by construction. The first property follows from Proposition 4.3. The loop spaces M := IBM and TV := IBN are in the genus of G respectively of L and give therefore rise to quasi-finite loop spaces. This finishes the proof. D Next we want to analyze the loop Qf3 : I/Q — » I/A of the gluing map /?. Lemma 5.2. Q/3 ~ j : LbQ ^ I/A. Proof. Using the H-space structure of I/A ~ QIC{4}A x IC(4)A, we can again describe the map fi/3 as a product of two maps, We just have to multiply the loops of /?i and /?2- Therefore, we only have to show that fi/?2 = Qf2°t(sl)~l) - QK{4}Q -^ QK(4)A is null homotopic. But this map fits into a commutative diagram  Vol. 74 (1999) The finiteness obstruction for loop spaces 667 where K' := QK{4) and K" := 1K"(4) are again Eilenberg-MacLane spaces. Since the first two arrows in the row are homotopy equivalences and since the map from the last column to the third column is the coaugmentation of the localization at 0, we only have to show that Qfzq : GA — * K"^ is null homotopic. The Eilenberg- MacLane space K(4)A ~ Yli ^(ZA, 2/) splits into a finite product such that I > 2 for all I. Since fP(£>G;ZA) is generated by classes of degree 4, each coordinate of the map >GA — » K(4)A is given by a decomposable cohomology class x\ in H*(BGA; ZA). The next lemma finishes the proof. D Lemma 5.3. Let L be a quasi-finite loop space and let g : >I/A — » K(ZA,r) denote a decomposable cohomology class in H*(BLA; ZA). Then, the map Qg : LA ~ ftBLA — > K(ZA, r - 1) 2S null homotopic. Proof. The adjoint of tig factors through Q5LA ^ BL ^ K(ZA,r). Since EQ5L is a co-H-space, all cup products vanish. Hence, the composition is null homotopic as well as lg. D 6. Proof of Theorem 1.4 We start with analyzing the genus of the connected compact Lie group G. Proposition 6.1. Let G be a semi simple connected compact Lie group, such that G = SU(2)r. Then, the genus of G is rigid. That is every element of the genus of G is actually homotopy equivalent to G. Proof. First we prove the statement for G = G = SU(2)r. Since GA ~ ^((Q (8) ZA)r,3) is an Eilenberg-MacLane space, the gluing map for M is given by a matrix A G G/(r, Q (8) ZA) and therefore splits into a product RC with R e G/(r,Q) and G G G/(r,ZA) [32]. Since G ^ SU(2)r, R can be realized by a self map G0 — > G0 and G by a self map GA — > GA. This implies that M ~ G [32]. Now let G be a quotient of G, and let M G genus(G). Then, 7Ti(M) = 7T1(G) =: E is a finite elementary abelian 2-group. The universal cover M is contained in the genus of G ~ SU(2)r. Hence, M ~ G and M fits into a fibration G -^ M -^ 5^. The 2-adic completion establishes an injection ^(G) -^ 7T3(GA) and turns the fibration into a principal fibration. Since principal fibration with  668 D. Notbohm CMH connected structure groups are simple and since self maps of G are classified by means of homotopy groups, the above fibration is oriented and is classified by a map BE — > BSHE(G), where SHE(G) denotes the set of all homotopy equivalence classes of G homotopic to the identity. The left action of G on itself establishes a map BG -> BSHE(G) [27]. Now, we consider the diagram BG > BSHE(G) Since G is a topological group, SHE(G^) is 2-complete and the map SHE(G)\ — > SHE(G\) is a homotopy equivalence [2]. Since completion and passing to classify ing spaces commutes for nilpotent connected loop spaces [2], the second arrow in the bottom line is a homotopy equivalence. The homotopy fibers of both columns have uniquely 2-divisible homotopy groups. Thus, obstruction theory shows that for every map BE — > BG% or BE — > BSHE(G%) there exist a lift to BG re spectively to BSHE(G), unique up to homotopy. A small diagram chase shows that, up to homotopy, the classifying map BE — > BSHE(G) has a unique lift BE — > BG. Hence, G — » M — > BE is a principal fibration. By [10], the clas sifying map is induced from a homomorphism p : E — » G. In fact, this map is a monomorphism, since the homotopy fiber M of Bp is quasi-finite [26]. All homomorphisms E — » G are central and therefore, M ~ G/ E is a compact Lie group. Finally we have to show that G/ E ~ G. Using shift maps and permutation of factors, one can show, analogously as for the equivalence SO (4) = SC/(2) xz/2 SC7(2) ~ SO(3) x SZ7(2), that G/E and G are homotopy equivalent to SO(3)S x SUi(2Y~s where s equals the rank of E. D Now we are able to finish the proof of Theorem 1.4. Proof of Theorem 1.4- The fibration L — » E — > BM of Proposition 5.1 has almost all the properties we stated in Theorem 1.4. It is only left to show that TV ~ L and that M ~ G. The first assertion follows from Lemma 5.2, and the second from Proposition 6.1. This finishes the proof of Theorem 1.4 D References [1] J. F. Adams & Z. Mahmud, Maps between classifying spaces, Inventiones Math. 35 (1976), 1-41. [2] A. Bousfield & D. Kan, Homotopy limits, completion and localizations, SLNM 304, Springer Verlag 1972.  Vol. 74 (1999) The finiteness obstruction for loop spaces 669 [3] W. Browder, Higher torsion in #-sspaces, Trans. AMS 108 (1963), 353-375. [4] W. G. Dwyer, H. Miller, C.W. Wilkerson, The homotopy uniqueness of BS3, Algebraic Topology Barcelona 1986, SLNM 1298, Springer, Berlin 1987, pp. 90-105. [5] W. G. Dwyer & C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. Math. (2) 139 (1994), 395-442. [6] W. G. Dwyer & C. W. Wilkerson, The center of a p-compact group, The Cech centennial (Boston, MA, 1993), Contemp. Math. 181 (1995), 119-157. [7] W. G. Dwyer & C. W. Wilkerson, Product splittings of p-compact groups, Fund. Math. 147 (1995), 279-300. [9] W. G. Dwyer & C. W. Wilkerson, p-compact groups with abelian Weyl groups, Preprint. [10] W. G. Dwyer & A. Zabrodsky, Maps between classifying spaces. In: Algebraic Topology, Barcelona 1986, SLNM 1298, pp. 106-11. [11] K. Ehrlich, Fibrations and transfer map in algebraic K-theory, JPPA 14 (1979), 131-136. [12] K. Ehrlich, The obstruction to the finiteness of the total space of a fibration, Michigan Math. J. 28 (1981), 19-38. [13] W. Luck, The transfer maps induced in the algebraic KQ- and K\ -groups I, Math. Scand. 59 (1986) 93-121. [14] W. Luck, The transfer map induced in the algebraic KQ- and K\ -groups by a fibration II, JPPA 45 (1987) 143-169. [15] G. Mislin, Wall's obstruction for nilpotent spaces, Topology 14 (1975), 311-317. [16] G. Mislin, Finitely dominated nilpotent spaces, Ann. Math. 103 (1976), 547-556. [17] G. Mislin, Wall's finiteness obstruction. In: Handbook of Algebraic topology (I. James, ed.), Elsevier 1995, pp. 1259-1292. [18] J. M. M011er, Rational isomorphisms of p-compact groups, to appear. [19] J.M. M011er, Homotopy Lie groups, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 413-428. [20] J. M. M011er & D. Notbohm, Centers and finite coverings of finite loop spaces, J. reine u. angew. Math. 456 (1994), 99-133. [21] J. M. M011er & D. Notbohm, Finite loop spaces with maximal tori, Math. Gott. Heft 14 (1994). [22] H. J. Munkholm & E. K. Pedersen, Whitehead transfer for S^-bundles, an algebraic de scription, Comm. Math. Helv. (1981), 404-430. [23] D. Notbohm, Kernels of maps between classifying spaces, Israel J. Math. 87 (1994), 243- 256. [24] D. Notbohm, Classifying spaces of compact lie groups and finite loop spaces. In: Handbook of Algebraic topology (I. James, ed.), Elsevier 1995, pp. 1049-1094. [25] E.K. Pedersen, Universal geometric examples for transfer maps in algebraic K- and L- theory, JPPA 22 (1981), 179-192. [26] D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549-572. [27] J. Stasheff, A classification theorem for fiber spaces, Topology 2 (1963), 239-246. [28] D. Sullivan, Geometric Topology, Part I: localizations, periodicity and Galois theory, Mimeo graphed notes, MIT 1970. [29] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. Math. (2) 81 (1965), 55-69. [30] C. T. C. Wall, Finiteness conditions for CW-complexes II, Proc. Roy. Soc. London Ser. A 295 (1966), 129-139. [31] G. W. Whitehead, Elements of homotopy theory, Graduate Text in Mathematics 61, Springer, Berlin 1978. [32] C. W. Wilkerson, Applications of minimal simplicial groups, Topology 15 (1976), 111-130.  670 D. Notbohm CMH D. Notbohm Mathematisches Institut Bunssenstr. 3-5 D-37073 Gottingen Germany e-mail: notbohm@at.cfgauss.uni-math.gwdg.de (Received: March 25, 1999)