nn X ____- X ____- X; whereis the q-th power map. Thus ad[q] : X ____- nX has the form ad[q](x) = [x]q: F.R. Cohen, and R. Levi * * 9 On the other hand the n-fold loop of [q] can be computed in nX as follows. On 1X the single loops on [q] is given by sending a generator [x] to its q-th power [x]q. Thus on nX the n-fold loops on [q] is given by sending a generator c(w)[x* *] to_ c([q]w)[x]q, as claimed. * * |__| 4. The construction LnX The model below for nX is constructed in a very similar fashion to nX. Assume that the complex X is n-connected and reduced. Replace the functor F n-1 in the construction of nX by Kan's loops group functor G. Thus define LnX to be n-1GX. The generating set in this case is more complicated to describe. However, o* *nce this is done LnX has some similar features to those of nX which are described in a later section. As before, start with an analysis of the group L2X = GX. As a free group, GX is generated by the set X = X=s0(-1 X). In dimension n, L2X is given as the kernel of the homomorphism dG0: (GX)n+1 ____- (GX)n: Since dG0is an epimorphism with right inverse sG0and (GX)n is free, there is a * *group isomorphism (GX)n+1 ~=(L2X)n o (GX)n: The following definition is useful. Definition 4.1. An epimorphism f : F _____- G of free groups is called proper* * if it has a right inverse s satisfying the following property: there exist sets X* * freely generating F and Y freely generating G, such that s maps Y into a subset of X. If f : F _____- G is a proper epimorphism of free groups, then with the nota* *tion above, let A(X) denote the subset s(Y ) of X. Let B0(X) denote the pointed comp* *le- ment of A(X). Let B(X) denote the subset of F given by all elements x . sf(x)-1 for all x 2 B0(X). Lemma 4.2. Let f : F ____- G be a proper epimorphism of free groups with ke* *rnel K. Then, with the notation above, K is freely generated by the set B(X)Fr[A(X)]. Proof. Let ff : B0(X) _____- B(X) be the map of sets given by ff(x) = x . sf(x* *)-1. Then ff is an epimorphism by construction. Let x1; x2 2 B0(X) be such that ff(x* *1) = ff(x2). Then ff(x1) = x1sf(x1)-1 = x2sf(x2)-1 = ff(x2): Thus x-12x1 = s(f(x2)-1f(x1)). But the left hand side of the last equation is a* * word in the free group on B0(X), whereas the right hand side is a word in the free g* *roup on A(X). Since these two sets intersect only on the base point, it follows that* * both sides are trivial so x1 = x2 and ff is an isomorphism of sets. Now write X as A(X) _ B0(X). There is an isomorphism of sets from X to the subset A(X) _ B(X) of F taking any element of A(X) to itself and any element of B0(X) to B(X) via ff. It is immediate that A(X)_B(X) freely generates F because sf(x) is in the free subgroup generated by A(X). Thus there is a commutative 10 Combinatorial models for iterated loop spaces diagram with exact rows f 1 ____________-K __________________-F _______________-G ________-1 | | | | | | | Fr[1_ff]| 1| ?| ?| ss ?| A 1 ____-F r[B(X)Fr[A(X)]]___-F r[A(X) _ B(X)] ___-F r[A(X)] ____-1 Since the right two vertical arrows are isomorphisms so is the left one and the* * result_ follows. * * |__| Definition 4.3. Define an operator on GX by (w) = sG0dG0(w). Lemma 4.4. Let X be a simplicial set and let X denote the generating set f* *or GX as defined above. Then a point x 2 X is a fixed point of if and only if x = s* *1y for some y 2 X. Proof. Since sG0 is the unique multiplicative extension of s1, it is clear tha* *t each fixed point x of in X is in the image of sG0. But x considered as a word in G* *X is of length 1 (or otherwise is the base point). On the other hand (x) = sG0dG0x = s1(d1(x)d0(x)-1) = s1d1(x) . s1d0(x)-1 is of length at most 2. Hence one of the factors must be trivial and we have x * *= s1dix for i = 0 or i = 1. Conversely, if x = s1y then (x) = (sG0y) = sG0dG0sG0y = sG0y = x: * * __ * * |__| Let X be a 2-connected reduced complex. Let A0(X) denote the graded set given by the intersection of X with the fixed point set of on GX. Let B00(X) denote the graded set given the pointed complement of A0(X) in X . Let A(X) and B0(X) denote the double desuspension of A0(X) and B00(X) respectively. Then (GX)n+1 = F r(Xn+2 ) = F [A(X)n _ B0(X)n]: Let B(X) denote the graded set, which in dimension n is the subset of (GX)n+1 given by all elements of the form ff(x) = x(x)-1 for x 2 B0(X)n. Then (GX)n+1 is freely generated by A(X)n _ B(X)n. The next corollary follows from Lemma 4.2. Corollary 4.5. The group L2X is isomorphic as a graded group to the free group F r[B(X)Fr[A(X)]]. Thus in terms of a word in the free group on the simplices of X every n-dime* *nsional generator in the specified generating set for L2X has the form c(w)(ff(x)), whe* *re w is a word in the free group on A(X)n and x 2 B0(X)n. A more convenient description for the group L2X, analogous to the presentation of 2X given above, will now be derived. Notation 4.6. For x 2 B0(X)n, denote ff(x) by [x]. Then the generating set * *for L2X defined above consists of symbols of the form c(w)[x], where w 2 F r[A(X)]. Consider the behavior of ff with respect to the simplicial operators. Notice* * that dL2iand sL2iare the multiplicative extensions of di+2 and si+2 respectively. Eq* *uiva- lently these operators can be thought of as extensions of dGi+1and sGi+1respect* *ively. The following lemma is routine and the verification is left to the reader. F.R. Cohen, and R. Levi * * 11 Lemma 4.7. With the notation above, the function ff satisfies the following* * rela- tions 1. For i > 0, dL2i[x] = [di+2(x)]: 2. For i 0, sL2i[x] = [si+2(x)]. 3. dL20[x] = d2(x)d1(x)-1d0(x): For any x 2 X define fl(x) = d2(x)d1(x)-1d0(x). Then dL20[x] = fl(x). The operator fl will be of some significance. Its commutation properties with resp* *ect to the simplicial operators are recorded in the following lemma. Here too, the verification is easy and is left for the reader. Lemma 4.8. The function fl satisfies the following relations 1. dL2i(fl(x)) = fl(di+3(x)): 2. sL2i(fl(x)) = fl(si+3(x)): 3. fl(sL20(x)) = [x]: Notice that 1 and 2 in the previous lemma imply that the set of all elements* * of the form fl(x), x 2 X form a simplicial subset of L2(X). Furthermore, 3 implies that this simplicial subset contains all elements of the form [x] = ff(x). The function ff has a useful multiplicative property which is recorded next.* * For any word w 2 GX define ff(w) = w . (w)-1. By analogy to the case when w is of length 1 denote ff(w) by [w]. Lemma 4.9. Let w = x1x2 . .x.n2 GX. Then [w] = [x1] . c((x1)-1)([x2]) . .c.((x1x2 . .x.n-1)-1)([xn]): Proof. For w of length 1 there is nothing to prove. Let w = x1x2 . .x.n. Then ff(w) = x1(x2 . .x.n)(x2 . .x.n)-1(x1)-1 = x1ff(x2 . .x.n)(x1)-1 = ff(x1)c((x1)-1)(ff(x2 . .x.n* *)): * * __ The claim follows by induction. * * |__| Notice that it was not assumed in the lemma that the presentation of w is re* *duced. Thus as an immediate corollary one obtains that for every w 2 GX, 1 = ff(1) = ff(ww-1) = ff(w)c((w)-1)(ff(w-1)): So that ff(w)-1 = c((w)-1)(ff(w-1)) or equivalently ff(w-1) = c((w))(ff(w)-1). In the notation used above [w-1] = c((w))([w]-1): Theorem 4.10. Let X be a 2-connected reduced complex. Then (L2X)n is isomor- phic as a group to the quotient of the free group on symbols c(w)[x], with [x] * *2 B(X)n and w 2 F r(A(X)n), by the relations c(w)[x] = 1 if x = 1. The simplicial operators are given by the formulas 1. sL2i(c(w)[x]) = c(sGi+1(w))[si+2(x)] for i 0. 2. dL2i(c(w)[x]) = c(dGi+1(w))[di+2(x)] for i 1. 3. dL20([x]) = [d2x]c(d0x)([d1x]-1[d0x]) and dL20(c(w)[x]) = c(dG1w)(c([dG1w])(dL20[x])): Proof. The calculation of the generating set is the contents of Corollary 4.5.* * It remains to verify the simplicial structure of L2X. By the remarks above one has 12 Combinatorial models for iterated loop spaces L L di2[x] = [di+2x] for i > 0 and si2[x] = [si+2x] for i 0. Thus dL2i(c(w)[x]) = c(dGi+1w)[di+2x] for i > 0 and similarly sL2i(c(w)[x]) = c(sGi+1w)[si+2x] for i 0. This give 1 and 2 in the theorem. For the calculation of dL20first recall that dL20[x] = fl(x) = d2x(d1x)-1d0x: Then observe that [d2x] . c(d0x)([d1x]-1[d0x]) = d2x(d2x)-1 . c(d0x)((d1x(d1x)-1)-1d0x(d0x)-1) = d2x(d2x)-1(d0x)-1(d1x(d1x)-1)-1d0x = d2x(d2x)-1(d0x)-1(d1x)d1x-1d0x = d2x(dG1x)-1(dG0x)d1x-1d0x = d2(x)d1(x)-1d0(x) = fl(x): In our notation the equation above reads dL20[x] = [d2(x)]c(d0x)([d1x]-1[d0x]): More generally, notice that dL20(c(w)[x]) = c(dG1w)(dL20[x]) = c([dG1w]dG1(w))(dL20[x]) = c(dG1(w))(c([dG1w])(dL20[x]* *): * * __ * * |__| Notice that lemma 4.9 gives a way of writing [dG1w] in the theorem explicitl* *y in terms of the generators once w is specified as a word in GX. The precise formul* *ation is omitted. The statement of Theorem 4.10 does not generalize well to Lm X for m > 2 as * *the notation becomes quite cumbersome. Before proceeding a more uniform description for the generating set is given in the next theorem. Theorem 4.11. Let X be a 2-connected reduced complex. Then (L2X)n is isomor- phic as a group to the quotient of the free group on symbolswith w 2 (G* *X)n and x 2 Xn+2 subject to the relations = 1 if x = siy for some y 2 Xn+1 a* *nd i 2 {0; 1}. The simplicial operators are given by the formulas 1. sL2i = for i 0. 2. dL2i = for i 1. 3. dL20<1; x> = <1; d2x> -1 and dL20 = c(w)(dL20<1; x>) Proof. Let U2X denote the graded free group generated in dimension n by the sy* *m- bols described in the theorem subject to the given relations. Then there is a m* *ap OE : U2X ____- L2X F.R. Cohen, and R. Levi * * 13 G sending to c(s0 w)[x]. To check that OE is well defined notice that if x = s0y then (x) = 1 and so [x] = x = s1y = 1 in GX. Hence generators of the form with x = s0y are in the kernel of OE. On the other hand if x = s1y for some y then [x] = 1 in GX since in this case (x) = x. Consequently OE is well defined. Furthermore, sin* *ce A(X) is isomorphic as a set to GX via the degeneracy operator sG0, it follows t* *hat OE is an epimorphism. To prove that OE is an isomorphism it remains to show th* *at its kernel vanishes. Thus it suffices to show that if w 2 (GX)n and x 2 Xn+2 th* *en c(sG0w)[x] = 1 if and only if x = siy for some y and i 2 {0; 1}. Indeed c(sG0w)[x] = 1 if and only if [x] = 1 as an element of GX. But [x] = x(x)-1. Thus [x] = 1 if and only if x = (x). By Lemma 4.4 this holds for x 2 X if and only if x = s1y for some y, whence the result. The simplicial operators can be calculated directly from Theorem 4.10 using * *the identification above. Notice that dL20 is not given in terms of the gene* *rating set. However, one can work out the explicit expression using Lemma 4.9 and the* *__ observation that w = [w](w). |* *__| The next theorem generalizes Theorem 4.11 to n 2. Theorem 4.12. Let X be an m-connected reduced complex for m 2. Then (Lm X)n is isomorphic as a group to the quotient of the free group on symbols ; x 2 Xn+m and wi 2 (LiX)n+m-i-1 ; subject to the relations = 1 if either 1. for some s < m the element represents the identity in L* *sX or 2. there exists a generator in Lm-1 X such that sLm-10 = : The simplicial operators are defined recursively as follows. 1. sLmi = for each i 0. 2. dLmi = for each i 1. 3. The face operator dLm0 can be expanded recursively using the formula dLm0 = c(wm-1 )(dLm-11(u).dLm-10(u)-1), where u* * denotes the generator of Lm-1 X. Proof. The statement of the theorem for m = 2 is the contents of Theorem 4.11. For u 2 LrX define r(u) = sLr0dLr0(u). For m > 2 assume by induction that the theorem holds for m - 1. For each 1 s m - 1 let GsX denote the subset of LsX given by the symbols ; x 2 Xn+s and wi 2 (LiX)n+s-i-1: Then by induction hypothesis GsX is closed under sLs0. Let RsX denote the subset of GsX given by 8 9 < | 9 2 Gs-1X; = | sLs-10 = * *or: : ; | = 1 inLjX forsome j < s - 1 14 Combinatorial models for iterated loop spaces Let Gs X denote the quotient set GsX=RsX. Then by induction hypothesis LsX is freely generated in the pointed sense by Gs X, for each s m - 1. Consider the projection dLm-10: P Lm-1 X ____- Lm-1 X; whose kernel is by definition Lm X. The formula for sLm-10 gives that dLm-10 i* *s a proper epimorphism of free groups with right inverse given by sLm-10. Write Gm-1 X = Am (X) _ B0m(X); where Am (X) = sLm-10(Gm-1 X) and B0m(X) = Gm-1 X=sLm-10(Gm-1 X): Let Bm (X) = {[u] := u . m-1 (u)-1 | u 2 B0m(X)} Lm X: Then by Lemma 4.2 there is an isomorphism Lm X ~=F r[Bm (X)Fr[Am (X)]]: Notice that the group F r[Am (X)] is isomorphic with the appropriate dimension * *shift to Lm-1 X. It is easy to observe, as in the proof of Lemma 4.2, that the correspondence u 7! [u] := u . m-1 (u)-1 is 1 - 1. Furthermore, if u 2 B0m(X), then [u] = 1 in* * Lm X if and only if u = 1 in Lm-1 X. To see this notice that if [u] = 1 then u = m-1* * (u). Hence in this case u is in the image of sLm-10, which means that it is trivial * *as an element of B0m(X). Finally notice that B0m(X) = Gm-1 X=sLm-10(Gm-1 X) = Gm-1 X=sLm-10(Gm-1 X) [ Rm-1 : Putting all this together, the presentation of Lm X follows. Notice that the symbol denotes the element c(sLm-10wm-1 )([ ]) in P Lm-1 X, where [u] := um-1 (u)-1. Thus for every i 0 one has sLmi = sLm-1i+1(c(sLm-10wm-1 )[ ]) = c(sLm-10sLm-1iwm-1 )[sLm-1i+1 ]; and the formula for the degeneracy operators follows by induction. The formula * *for the face operators dLmi for i 1 follows in a similar fashion. Finally let u = . Then we have dLm0 = dLm-11(c(sLm-10wm-1 )(u . m-1 (u)-1) = c(wm-1 )(dLm-11(u) . dLm-10(u)-* *1): * * __ This gives the recursive formula for dLm0 and thus completes the proof. * * |__| Example 4.13. The calculation of dL30on a typical generator of the form is given next. By the theorem one has dL30 = c(w2)(dL21 . dL20 -1) = c(w2)( . (c(w1)(dL20<1; x>-1* *))): Expanding this further can be done by using the identities wi = [wi]i(wi), where in each case [wi] means wii(wi)-1. In each case one needs to know an explicit F.R. Cohen, and R. Levi * * 15 decomposition of wi as a word in its respective group. The simplest instance of* * this occurs for wi = 1 for i = 1; 2. In this case one has dL30<1; 1; x> = <1; 1; d3x>(<1; 1; d2x><1; dG0d0x; d1x>-1<1; dG0d0x; d0* *x>)-1: 5. Comparison of the models Recall that nX = n-1F n-1X, whereas LnX = n-1GX. The following theorem summarizes some obvious relationships between the two constructions. Proposition 5.1. Let Y be connected then 1. If X = nY then LnX = nY . 2. If Y = LnZ then Y is a retract of nY via explicit maps En Lnevn LnZ ____- nLnZ _____- LnZ: Proof. The first statement follows at once from the definitions and the observ* *ation that the functors G and F are naturally isomorphic. For the second statement recall that the k-fold Freudenthal suspension map w* *as described earlier. There is a natural way to write the simplicial analogue of * *the map nevn. The construction becomes a triviality once it is observed that there * *is a natural map ev : W _____- W for every simplicial set W (categorically this is just the fact that is left adjoint to in the category of simplicial sets). On* *ce this map is constructed, apply its (n - 2)-fold suspension to W = n-2GZ to get a map n-2ev : n-1n-1GZ ____- n-2n-2GZ: Inductively one gets a map evn-1 : n-1n-1GZ ____- GZ: The target being a simplicial group this map extends to a simplicial homomorphi* *sm ad(evn-1) : F n-1n-1GZ ____- GZ: This simplicial homomorphism can now be looped n - 1 times to give the desired simplicial homomorphism nevn : nLnZ ____- LnZ: Thus define ev : Y _____- Y as follows. Let (y; j) 2 (Y )n be any element. Hence y is a simplex of dimension n - j + 1 in Y with the additional property t* *hat d0(y) = *. Define ev(y; j) = sj-10(y): It is a routine verification that this map is simplicial, and natural in Y . * * __ * * |__| For an explicit example, consider 2ev2 : 2L2X _____- L2X. Thus start with ev : L2X ____- GX. A typical element has the form (w; j) and is sent by definit* *ion to (sG0)j-1w under ev. In particular an element of the form [w] = (w; 1) is sen* *t to w. A generator for 2L2X has the form c((u1; i1) . .(.uk; ik))[w]. Thus 2ev2 is * *the unique multiplicative extension of the map sending an element of the form above* * to c((sG0)i1-1(u1) . .(.sG0)ik-1(uk))(w): Notice that for any simplicial set X, the construction of the Moore loops X makes no use of the simplicial operator s0. Thus X has an extra operator which 16 Combinatorial models for iterated loop spaces we may denote by s-1. Notice however that the use of the word "operator" here is misleading. Namely, if x 2 X is a non-trivial simplex such that d0x is t* *he base point so that x is in fact a simplex in X, then s-1x is not in X, since d0 s-1x = d1s0x = x. To fix this one may think about s-1 as a function from X to X, which is degree preserving. Notice that an i-fold iteration of s-1 may be th* *ought of as a function of degree i - 1 from X to X. For every i 0 and n 2 define an operator of degree -1 OEi : LnX ____- Ln-1X by OEi = (sLn-1)i(dLn0)i; where (sLn-1)i stands for (sLn-10)i considered as an operator from Ln to Ln-1. * *Notice that if X is n-connected and reduced and w 2 (LnX)k then OEk(w) = 1. Notice also that the operators OEi defined above for LnX are inherited by nX by 1 of Proposition 5.1. The following lemma summarizes the commutation features of OEi with the simplicial operators. Lemma 5.2. Let for i 1 and n 2, let OEi : LnX ____- Ln-1X denote the oper* *ator defined above. Then 1. dLn-1jOEi = OEi-1dLnjif j i 2. dLn-1jOEi = OEidLnj-1if j > i 3. sLn-1jOEi = OEi+1sLnjif j i 4. sLn-1jOEi = OEisLnj-1if j > i. Proof. We verify 1 and 3. The other two identities follow by analogy and are l* *eft to the reader. Notice that dLn-1jOEi = dLn-1j(sLn-10)i(dLn-11)i: One then uses the simplicial identities to verify that for j < i djsi0di1= si-10di-11dj+1: Back in LnX the right hand side is OEidLnj. This gives 1. To verify 3 notice th* *at the left hand side is sjsi0di1= si+10di1: On the other hand the right hand side is si+10di+11sj+1 = si+10di-k1sj-kdk+11= si+10di-j+11s1dj1= si+10di-j1dj1= * *si+10di1 * * __ implying 3. * *|__| The operators OEi become useful when one writes explicit homotopies on these models. The idea of how these are used is demonstrated below in writing an expl* *icit null-homotopy for the commutator map on an iterated loop space. The standard 1-simplex [1] is the simplicial set which in dimension n has n * *+ 2 simplices denoted <0i; 1n+1-i> for 0 i n + 1. Faces and degeneracies are giv* *en by the formulas ( <0i-1; 1n+1-i>j < i dj<0i; 1n+1-i> = <0i; 1n-i> j i F.R. Cohen, and R. Levi * * 17 ( <0i+1; 1n+1-i>j < i sj<0i; 1n+1-i> = <0i; 1n+2-i> j i If h : [1] x X _____- Y is a homotopy (i.e. a simplicial map), denote by hi t* *he function defined for xn 2 Xn by hi(xn) = h(<0i; 1n+1-i>; xn): Notice that hi is only defined on xn with n i - 1. Let X be an n-connected reduced complex, n 2, and let G denote LnX. Define o : [1] x G x G ____- G by oi(u; v) = [u; OEi(v)]; for every pair of words u; v 2 G. Lemma 5.3. The map o defined above is a simplicial null-homotopy for the co* *m- mutator map. Proof. For u; v 2 Gk one evidently has ok+1(u; v) = [u; 1] = 1 and o0(u; v) = [u; v]: The verification that o is a simplicial map is a standard exercise, using the n* *otation given above for the 1-simplex, Lemma 5.2 and the simplicial identities, is left* * to_the reader. * *|__| 6. Filtrations for n and Ln There is a filtration of nX by simplicial subgroups obtained as follows. Re* *call that nX is freely generated by elements of the form of the form c(w)[x]. The wo* *rd w is an element in a free group F r[An(X)] and thus has the usual reduced word length filtration. Give c(w)[x] filtration k if w has filtration k in the word* * length filtration. Let Fj2X be the subgroup of nX generated by all c(w)[x] of filtrati* *on at most j. Lemma 6.1. The simplicial groups FjnX endow nX with the structure of a fil- tered simplicial group with F0nX isomorphic to F X. Proof. First notice that the simplicial operators preserve the generating set * *of the group FjnX. This ensures that each FjnX is indeed a simplicial subgroup of nX. Filtration zero, F0nX, is exactly the subgroup generated by the image of_ the Freudenthal suspension En : X ____- nX. Thus F0nX = F X as claimed. |__| The filtration on LnX arises in a similar way, but describing it requires so* *me preparation. First, let X be a simplicial set and let Y be a subset, which is* * not necessarily simplicial. Define the simplicial envelope of Y in X, EnvX (Y ) to * *be the simplicial subset of X given by the intersection of all simplicial subsets cont* *aining Y . Thus EnvX (Y ) is the minimal simplicial subset of X containing Y . Obvious* *ly EnvX (Y ) is obtained from Y by adding in all iterations of faces and degenerac* *ies of simplices in Y . If G is a simplicial group and X a subset of G, then the simplicial closure * *EnvG (X) is not necessarily a simplicial subgroup. However the following holds. 18 Combinatorial models for iterated loop spaces Lemma 6.2. Let G be a simplicial group and let X be a subset such that EnvG* * (X) is obtained from X by adjoining to X all elements of G of the form di0(x), x 2 * *X for some i 0. Let denote the subgroup of G generated by X. Then EnvG ( ) is a simplicial subgroup of G. Moreover EnvG ( ) = : Proof. Since EnvG (X) contains X, the subgroup contains . By definition EnvG ( ) is the intersection of all simplicial subsets of G contai* *ning . Hence, EnvG ( ). Next notice that EnvG (X) is obtained from X by adjoining all elements of the form d0(y), y 2 X. This follows at once from the identity dk0= d0d1 . .d.k-1and the assumption that X is closed under a* *ll simplicial operators except possibly d0. Thus every element of can * *be written in the form w = xffl11d0(y1)ffi1.x.f.flrrd0(yr)ffir; where xi; yj 2 X and ffli; ffij 2 {-1; 0; 1}. But xffl11d0(y1)ffi1.x.f.flrrd0(yr)ffir= d0(s0(x1)ffl1d0(y1)ffi1.s.0.(xr)* *fflrd0(yr)ffir) and the right hand side is the image under d0 of an element in . This gives * *and_ inclusion the other way and the proof is complete. * * |__| We are now ready to define the filtration for LkX. The group LkX is freely generated by symbols with wj 2 LjX. Let TrLkX denote the subset of the generating set consisting of all generators of the form above* * such that wk-1 is a word of reduced length at most r. Notice that TrLkX is closed under all simplicial operators with the exception of dLk0. Thus the simplicial * *subset EnvLkX (TrLkX) is obtained from TrLkX by adjoining all the images of its elemen* *ts under dLk0. Consequently EnvLkX ( ) is a simplicial subgroup of LkX and is in fact equal to the simplicial subgroup generated by EnvLkX (TrLkX). Define FpLkX = EnvLkX ( ): 7. A simplicial model for Map*(Sn;kg; X) Let g be a non-negative integer and let Sg denote the closed Riemann surface* * of genus g. If k is another non-negative integer, let Skgdenote a Riemann surface* * of genus g and k distinct boundary components. In particular S0gis identified with* * Sg. Let X be an arbitrary space. In various applications one is interested in the s* *pace of maps from Skgto X. In this section simplicial models for Map*(Skg; X) will * *be constructed, where here X denote an arbitrary simplicial set. The intuition for this construction comes from the, rather simple, homotopy * *the- ory of these spaces. First notice that up to homotopy, a Riemann surface with k > 0 boundary components, is equivalent to a singular surface whose boundary consists of a wedge of k circles. Furthermore, as a cell complex the surface Sk* *gcan be constructed by means of the cofibration sequence ffkg _g 1 1 _k 1 k S1z____- { (Sxi_ Syi)} _ Sbj ____- Sg; i=1 j=1 where subscripts on the circles denote the generators of the respective fundame* *ntal groups and the map ffkgis specified up to homotopy by its effect on fundamental F.R. Cohen, and R. Levi * * 19 groups as ffkg= [x1; y1][x2; y2] . .[.xg; yg] . b-1j. .b.-11: For an arbitrary pointed space X, the mapping space functor Map*(-; X) turns the cofibration above into a fibration and Map*(Skg; X) is defined as the fibre spa* *ce in this fibration, where the projection map is induced by ffkg. For non-negative integers g and k consider the system akg d0 (GX)2g+k ____- GX oe___ P GX: The map akgis given as follows akg(u1; v1; . .;.ug; vg; z1; . .;.zk) = [u1; v1][u2; v2] . .[.ug; vg](z* *1 . .z.k)-1: Let Mkg(X) denote the pull-back of this system. Theorem 7.1. If X is a simply-connected reduced complex then Mkg(X) is a Kan complex and its geometric realization is homotopy equivalent to Map*(Skg; |X|).* * The simplicial operators in Mkg(X) are induced from those of GX and P GX. Proof. First notice that Mkg(X) is a Kan complex as a pull-back space in a dia* *gram of Kan complexes. That the maps akgmodel the geometric maps induced by ffkgis obvious by construction. Notice also that since the projection d0 : P GX ____- * *GX is a Kan fibration, the homotopy type of Mkg(X) is a homotopy invariant of X. T* *his_ completes the proof. * *|__| Theorem 1.5 follows at once from Theorem 7.1. In particular for g = k = 0 the bottom left hand side corner of the pull-back square above is trivial and so M0* *0(X) is simply the kernel of dG0, namely the model L2(X). Notice that in general Mkg(X) has a natural action of the simplicial group L* *2X, with homotopy orbit space equivalent to (GX)2g+k. 8. Appendix This appendix is dedicated to the proof of the following lemma, which is well known but we are not aware of an appropriate reference. Lemma 8.1. Let A and B be free groups generated by sets X and Y respective* *ly. Then the kernel of the natural projection ss : A*B ____- A is the free group ge* *nerated by the set Y A, namely by all conjugates of elements in Y by words in A. Proof. Let A, and B be free groups. Let C"denote a tree on which A acts proper* *ly discontinuously with the orbit space C"=A = C. Similarly, let D" denote a tree* * on which B acts properly discontinuously with the orbit space "D=B = D. Without lo* *ss of generality we may label the edges of C and D by X and Y respectively. Then C" has one vertex for any element in A and one edge between two elements if the* *ir difference is a member of the generating set X. Similarly for D". Fix a base-point in D, which is the image of any vertex in "D, and attach on* *e copy of D to each vertex in C" by identifying each vertex of C" with this fixed base* *-point in D. Call the resulting adjunction space E. Then the action of A on C" descends to a properly discontinuous action of A * *on E with quotient C _ D. Thus there is a split short exact sequence of groups 1 ! ss1(E) ! ss1(C _ D) ! A ! 1: 20 Combinatorial models for iterated loop spaces Notice that the fundamental group ss1(E) is a free group which has basis given * *by the ordered pairs (ff; fi) where ff runs over all vertices of "Cand fi runs ove* *r all edges of D (indexed by Y ). A choice of basis is gotten as follows: Choose a path from the vertex ff to* * the base-point. The basis element (ff; fi) is represented the class on the loop ff-* *1 . fi . ff in ss1(E) by standard covering space theory. The image of ss1(E) ! ss1(C _ D) is * *thus__ given precisely by the subgroup freely generated by Y A. The results follows. * * |__| References [1]E. Curtis, Simplicial Homotopy Theory, Advances in Math. 6 1971 107-209. [2]D. Kan, A combinatorial description of homotopy groups, Ann. of Math., Vo* *l. 67, No. 2, (1957) 282-312. [3]W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Dover (* *1976) [4]J. P. May, Simplicial objects in algebraic topology, Van Nostrand Mathema* *tical Studies #11 (1967) [5]J. Milnor, The construction F K, Princeton, (1956) (mimeographed notes) [6]J. C. Moore, Seminar on Algebraic homotopy theory, unpublished lecture no* *tes. [7]J. Smith, Simplicial group models for nSnX, Israel J. Math. 66 (1989), no* *. 1-3, 330-350.