MAPPING SPACES AND HOMOLOGY ISOMORPHISMS NICHOLAS J. KUHN Abstract. Let Map (K, X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E* be a gen- eralized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map (K, X) will send a E*-isomorphism in either variable to a map that is monic in E* ho- mology. Interesting examples arise by letting E* be K-theory, K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces. 1.Introduction and main results Let K and X be pointed spaces, with K homotopy equivalent to a finite cell complex, and then let Map (K, X) denote the space of pointed continuous maps from K to X. Fixing K, this includes many important constructions on X. For example, Map (Sn, X) = nX, the nth loopspace of X, and Map (S1+, X) = LX, the free loopspace on X. Suppose E* is a generalized homology theory. A fundamental problem is to try to determine to what extent E*(Map (K, X)) might be determined by E*(X). This is difficult, and has a long history, even when E* is ordinary homology with field coefficients and K = S1. We consider a related problem. A map f : X ! Y will be called an E*-isomorphism if E*(f) is an isomorphism. One can ask to what extent does Map (K, ) preserve E*-isomorphisms? This is question of interest when E* is a nonconnective theory as the following simple example illustrates: the constant map c : K(Z=p, 2) ! * is an isomorphism in complex K-theory K*, but c : K(Z=p, 1) ! * is not. A much more subtle family of examples has been constructed recently by L. Langsetmo and D. Stanley [LS ]: see Example 1.4 below. In this paper we use Goodwillie calculus methods to prove the curious result that under suitable conditions on K and X, Map (K, X) will send an E*-isomorphism in either variable to a map that is monic in E* homology. 1.1. The main theorem. We need to define some numerical invariants of spaces. ____________ Date: July 8, 2004. 2000 Mathematics Subject Classification. Primary 55P35; Secondary 55N20, 55P* *42. This research was partially supported by a grant from the National Science F* *oundation. 1 2 N.J.KUHN Let d(K) be the minimal d such that K is homotopy equivalent to a d-dimensional complex. Let e(K) be the minimal n such that there exists a parallelizable n- dimensional manifold M, together with a closed subcomplex A such that K is homotopy equivalent to M=A. For example, e(Sn) = n, as Sn = Dn=Sn-1 . Let c(X) be the connectivity of X. Let s(X) be the minimal n such that X is homotopy equivalent to an n-fold suspension. Armed with these definitions, we can state our main theorem. Theorem 1.1. Suppose e(K) s(X) and d(K) c(X). (1) If f : X ! Y is an E*-isomorphism, then Map (K, f) : Map (K, X) ! Map (K, Y ) is E*-monic. (2) If g : L ! K is an E*-isomorphism of finite complexes, then Map (g, X) : Map (K, X) ! Map (L, X) is E*-monic. Corollary 1.2. If Z is connected, and f : nZ ! Y is an E*-isomorphism, then nf : n nZ ! nY is E*-monic. Remark 1.3. This corollary seems to be new even when n = 1. To the best of the author's knowledge, the only results of this sort in the literature are the author's papers [K1 , K2 ] which contain the n = 1 version of the corollary. Note that d(K) e(K). Furthermore s(X) c(X)+1 is always true, and very often s(X) c(X). For example, s(Mn (d)) = c(Mn (d)) = n - 2 where Mn (d) is the Moore space Dn [d Sn-1 . Thus when the first inequality in the hypotheses of the theorem holds, so usually does the second. In general, e(K) seems hard to compute exactly. The appendix includes some observations of Greg Arone and the author which yield some further explicit calculations, and some general bounds. For example, e(Mn (d)) = n+1, and, e(K) 2d(k) - 1 for all K with d(K) 1. The numeric hypotheses of our theorem are easy to explain. The condition d(K) c(X) guarantees the strong convergence of the Goodwillie tower of the functor sending a space X to the suspension spectrum 1 Map (K, X). The condition e(K) s(X) implies that there is a filtered configuration space approximation to Map (K, X), as in work of Bödigheimer [Bö ], follow- ing McDuff [McD ] and May [Ma ]. When both numeric conditions hold, statement (1) of Theorem 1.1 is proved by using properties of Goodwillie towers to play the two correspond- ing geometric conditions against each other. Using the existence of Bousfield localization of spaces, statement (2) is then a formal consequence of (1). MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 3 The proof of Theorem 1.1 is given in sections 2 and 3. In section 2, we outline how general calculus theory leads to theorems like ours, while in the shorter section 3, we specialize to the case in hand. 1.2. Examples and applications. Example 1.4. Let p be an odd prime. For each (m, n) in an explicit infinite list of pairs, with m 4 and both m and (n - m) taking on arbitrarily large values, Langsetmo and Stanley [LS ] construct a K*-isomorphism f : Mn (p) ! Mm (p) such that f is not a K*-isomorphism. For example, with p = 3, for all t 1, one has such a nondurable K*-isomorphism f : M4t(3) ! M4(3). It is not hard to deduce that then, for all j 1, the 3-connected cover of jf, ( jf)<3> : jMn (p)<3> ! jMm (p)<3>, is also not a K*-isomorphism1. In contrast, Corollary 1.2 implies that for all 1 j n - 2, jf : jMn (p) ! jMm (p) is K*-monic. Combining these results, we conclude that for 1 j n - 5, ( jf)<3> : jMn (p) ! jMm (p)<3> is K*-monic but not K*-epic. Example 1.5. When the homology theory E* is K(r)*, the rth Morava K-theory at a prime p, the corollary has the following computational im- plication. Let f : nZ ! Y be a K(r)*-isomorphism, with Z connected and n 1, and let F be the fiber of f. The K(r)* bar spectral sequence associated to the principal fibration nf n nZ --! nY ! n-1F converges to K(r)*( n-1F ) and has n nZ) n E2*,*= TorK(r)*(*,* (K(r)*( Y ), K(r)*). By the corollary, ( nf)* n K(r)*( n nZ) ----! K(r)*( Y ) is monic. The map ( nf)* is in the category of K=p-Hopf algebras studied by Bousfield in [B2 , Appendix]. He shows [B2 , Thm.10.8] that objects in ____________ 1This follows from a theorem of Bousfield [B3, Theorem 11.10], but is easy t* *o prove directly, using that ~K*(K(Z=p, 2) = 0 4 N.J.KUHN this category are flat over subobjects, when viewed as algebras. We conclude that the spectral sequence collapses, giving an isomorphism K(r)*( n-1F ) ' K(r)*( nY ) K(r)*( n nZ)K(r)* of K(r)*-coalgebras2. Example 1.6. Suppose g : L ! K is a K(r)*-isomorphism between finite complexes. Let C be the cofiber of g. Applying statement (2) of Theorem 1.1 to g, and reasoning as in the last example, we deduce that, for all X such that e(K) < s(X) and d(K) < c(X), one gets an isomorphism of K(r)*- coalgebras K(r)*(Map (C, X)) ' K(r)*(Map ( L, X)) K(r)*(Map( K,X))K(r)*. 1.3. Acknowledgements. The simple argument given in x3 proving that statement (1) of Theorem 1.1 implies statement (2) is due to Pete Bousfield, and replaces a different argument by the author, which needed the side hypothesis that E* be a ring theory. For this, and for other encouraging `e'- conversations, I offer Pete my thanks. Thanks are also due my colleagues Greg Arone and Slava Krushkal for discussions about this material. 2.Goodwillie calculus and E*-isomorphisms Let T denote the category of based spaces, and S a nice model category of spectra, e.g. the category of S-modules of [EKMM ]. In this section we find conditions on a functor F : T ! S and a space X ensuring that if f : X ! Y is an E*-isomorphism, then F (f) : F (X) ! F (Y ) will be E*-monic. 2.1. Review of Goodwillie calculus. In the series of papers [G1 , G2 , G3 ], Tom Goodwillie has developed his theory of polynomial resolutions of homotopy functors. We need to summarize some aspects of Goodwillie's work as they apply to functors F : T ! S. As carefully discussed in [G2 , G3 ], a functor is said to be polynomial of degree r if it takes strongly homotopy cocartesion (r + 1)-cubical diagrams to homotopy cartesian cubical diagrams. In [G3 ], given a functor F from one topological model category to another, Goodwillie proves the existence of a tower {PrF } under F so that F ! PrF is the universal arrow to a polynomial functor of degree r, up to weak equivalence. The functors F of interest to us in this paper satisfy an additional prop- erty: they will be finitary. Here, following [G3 , Definition 5.10], F is said * *to be finitary if it commutes with filtered homotopy colimits up to equivalence. Examination of the construction of PrF shows that Pr satisfies the fol- lowing useful properties. ____________ 2K(r)*-Hopf algebras if n > 1. MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 5 Lemma 2.1. (Compare with [G3 , Proposition 1.7].) (1) If F (X) ! G(X) ! H(X) is a fibration sequence for all X, so is PrF (X) ! PrG(X) ! PrH(X). (2) Given natural transformations F1 ! F2 ! . .,.the natural map hocolimsPrFs(X) ! Pr(hocolimsFs)(X) is an equivalence for all r and X. (3) If F is finitary, so is PrF for all r. The fact that the suspension of a strongly homotopy cocartesian cube is again strongly homotopy cocartesion implies the next property of Goodwillie towers. Lemma 2.2. [G3 , Remark 1.1] There is a natural equivalence Pr(F O d)(X) ' (PrF )( dX). Let DrF (X) be the homotopy fiber of PrF (X) ! Pr-1F (X). DrF is homogeneous of degree r: it has degree r, and Pr-1DrF (X) is weakly con- tractible. (This follows from Lemma 2.1(1): see [G3 , Proposition 1.17].) Goodwillie analyzes DrF . We need his description when F is also finitary and takes values in a stable model category like S. Proposition 2.3. [G3 , Theorems 3.5, 6.1] If F : T ! S is finitary, then, for each r, there is a spectrum tr(F ) with an action of the rth symmetric group r, and a natural weak equivalence DrF (X) ' (tr(F ) ^ X^r)h r. Corollary 2.4. If F : T ! S is finitary, and f : X ! Y is an E*- isomorphism, then PrF (f) : PrF (X) ! PrF (Y ) is also an E*-isomorphism. Proof.Standard spectral sequences show that any construction of the form (C ^X^r)h r preserves E*-isomorphisms in the X variable. The proposition thus implies that the maps on fibers, DrF (f) : DrF (X) ! DrF (Y ), are E*- isomorphisms. The proposition then follows by induction on r. Remark 2.5. The corollary is false without the finitary hypothesis. Examples can easily be constructed using homological localization functors, which are homogeneous and linear, but not, in general, finitary. 2.2. Strongly split towers of spectra. Suppose we are given a tower of spectra under another spectrum: 6 N.J.KUHN .. . | | fflffl| @C2@ p1|| e2 fflffl| p8C18 ppp pe1pppp p0|| pppp e0 fflffl| C ___________//C0. We will say that the tower is strongly convergent if the connectivity of the maps er goes to infinity as r goes to infinity. We will say that the tower is strongly split if there exists a homotopy commutative diagram .. .OO | | | C2 OO |j | 1 i2 | p C1 ppp OO i1ppp |j ppp | 0 xxpppi0 | C oo_________C0. such that, for all r, the composite Cr ir-!C er-!Cr is an equivalence. The following lemma is evident. Lemma 2.6. If a tower as above is both strongly convergent and strongly split then the induced map hocolimrir : hocolimrCr ! C is an equivalence. Thus, if E* is a homology theory, then colimrE*(Cr) ! E*(C) is an isomorphism. Remark 2.7. This lemma says all that we will need to know about strongly split towers for our purposes. However, it is illuminating to note the follow- ing. If a tower is strongly split, one can, if needed, modify the splitting data MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 7 so that er O ir : Cr ! Cr is homotopic to the identity. In this case, it will also true that the composite Cr jr-!Cr+1 pr-!Cr will be homotopic to the identity for all r. If the tower is also strongly convergent, then there will be a wedge decomposition 1` C ' hofiber{pr : Cr ! Cr-1}. r=1 2.3. A useful proposition. Proposition 2.8. Let F : T ! S be finitary. Suppose the Goodwillie tower of F is both strongly convergent and strongly split when evaluated at a space X. Then, if f : X ! Y is an E*-isomorphism, then F (f) : F (X) ! F (Y ) is E*-monic. Proof.Let ir : PrF (X) ! F (X) and jr : PrF (X) ! Pr+1F (X) denote the maps splitting the tower {PrF (X)}. Suppose f : X ! Y is an E*- isomorphism, and consider the diagram ir er PrF (X) __________//_F (X)_________//_PrF (X) | | | | |F(f)| PrF(f)|| | | | | fflffl| er fflffl| F (Y )__________//PrF (Y ). By assumption, the top composite is an equivalence, and thus an E*- isomorphism. Since f is an E*-isomorphism, so is the right vertical map, by Corollary 2.4. We conclude that E*(F (f)) is monic when restricted to the image of E*(ir). But E*(F (X)) is the colimit over r of these images, by Lemma 2.6, and thus E*(F (f)) is also monic. To apply this proposition, we need criteria ensuring that a Goodwillie tower {PrF (X)} strongly splits. This is the topic of our next two subsec- tions. 2.4. Goodwillie towers of functors with polynomial filtration. Say that a functor C : T ! S has a polynomial filtration if it is filtered by functors F0C ! F1C ! . .s.uch that hocolimrFrC(X) ! C(X) is an equivalence, and the homotopy cofiber functor FrC=Fr-1C 8 N.J.KUHN is homogeneous of degree r for all r. The following lemma is well known folk knowledge. Lemma 2.9. In this situation, the composite FrC(X) ! C(X) ! PrC(X) is an equivalence for all r and X. It follows that the Goodwillie tower {PrC(X)} will be strongly split. Proof.We have a homotopy commuative diagram with rows that are cofi- bration sequences FrC(X) _______//C(X)________//C=FrC(X) | | | | | | fflffl| fflffl| fflffl| PrFrC(X) _____//PrC(X)_____//Pr(C=FrC)(X). As FrC has degree r, the left vertical map is an equivalence. If we check that Pr(C=FrC)(X) ' *, then the bottom left map will be an equivalence, and we will be done. To check this we have Pr(C=FrC)(X) ' hocolimsPr(FsC=FrC)(X) ' *, as Pr(FsC=FrC)(X) ' * for s r. 2.5. Stable natural equivalences. Call a natural transformation (X) : C(X) ! G(X) a stable equivalence if it is an equivalence for all suitably connected spaces X. Lemma 2.10. If : C ! G is a stable equivalence, then Pr (X) : PrC(X) ! PrG(X) is an equivalence for all X. Proof.An examination of the construction of Pr shows that if (X) is an equivalence for all (d - 1)-connected spaces X, then so is Pr (X) : PrC(X) ! PrG(X). Now we can apply [G3 , Corollary 3.8] which implies that, for any functor F : T ! S, PrF is determined by its values on d-fold suspensions. Corollary 2.11. Suppose : C ! G is a stable equivalence. If, for a particular X, the Goodwillie tower {PrC(X)} is strongly split, then so also is the Goodwillie tower {PrG(X)}. 3. Proof of Theorem 1.1 The Goodwillie tower of the functor from spaces to spectra sending X to 1 Map (K, X) consists of a diagram of functors MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 9 .. . | | fflffl| P3K(X);; wwww | ww | e3 wwww fflffl| www k5P2K(X)5 www kkk ww e2kkkk | wwwkkkkk | wwkkk e fflffl| 1 Map (K, X) ___1______//_P1K(X). In [G2 , Example 4.5], it is shown that this tower is strongly convergent if d(K) c(X). Thus Theorem 1.1(1) will follow from Proposition 2.8 once we show that the tower is strongly split whenever e(K) s(X). Otherwise said, we wish to show that if n e(K), then the tower {PrK( nZ)} is strongly split for all spaces Z. By Lemma 2.2, this tower agrees with the tower associated to the functor sending a space Z to 1 Map (K, nZ). In the terminology of the last section, the main constructions and theo- rems of [Bö ] states that if n e(K), then there is a filtered configuration space C(K, Z) such that 1 C(K, Z) is a functor with a polynomial filtra- tion, and a natural map of spaces C(K, Z) ! Map (K, nZ) such that 1 C(K, Z) ! 1 Map (K, nZ) is a stable equivalence. Then Lemma 2.9 and Corollary 2.11 combine to say that the tower associated to 1 Map (K, nZ) is strongly split. Statement (2) of Theorem 1.1 turns out to follow easily from statement (1). The following argument was observed by Pete Bousfield. Suppose g : L ! K is an E*-isomorphism between finite complexes. Let X ! XE be Bousfield localization of the space X with respect to E* [B1 ]. Consider the diagram Map (K, X) ___________//Map(K, XE ) | | | | |Map(g,X)| |Map(g,XE)| | | | | fflffl| fflffl| Map (L, X) ___________//Map(L, XE ). As X ! XE is an E*-isomorphism, statement (1) of Theorem 1.1 applies to say that the top map is E*-monic. The right vertical map is a homotopy 10 N.J.KUHN equivalence as XE is E*-local, and is thus an E*-isomorphism. Thus the left vertical map is E*-monic. Remark 3.1. Though we haven't needed this here, there is an explicit model for the tower {PrK(X)} for 1 MapT (K, X): see [Ar , AK ]. From this model, it follows that a version of Corollary 2.4 holds for the K-variable: if E* is a ring theory, and g : L ! K is an E*-isomorphism, then so is Prf(X) : PrK(X) ! PrL(X). This leads to an alternative proof of Theorem 1.1(2), under the ring theory hypothesis. Appendix A. Computations of e(K) By Greg Arone and Nicholas Kuhn Recall that e(K) is the minimal n such that there exists a parallelizable n-dimensional manifold M, together with a closed subcomplex A such that K ' M=A. In this appendix we make some observations allowing for some general estimates and explicit computations of e(K). A.1. Upper bounds. If K ' M=A, with M parallelizable, we will say that the pair (M, A) represents K. Lemma A.1. e(K ^ L) e(K) + e(L). Proof.If (M, A) represents K and (N, B) represents L, then (M x N, A x N [ M x B) represents K ^ L. Corollary A.2. e( rK) e(K) + r. Lemma A.3. Suppose that L is a finite complex equivalent to a stably par- allelizable m-manifold. If n > m, and K is obtained from L by attaching n-cells, then e(K) m + n. Proof.Suppose g : L ! M is an equivalence, with M a stably parallelizable m-manifold. Let c a ac fi: Sn-1 ! L i=1 i=1 ` be the attaching maps for constructing K from L. Then` ci=1Sn-1 embeds in the parallelizable manifold DnxM so that (DnxM, ci=1Sn-1 ) represents K. Corollary A.4. If K is the mapping cone of a map f : Sn ! Sm , then e(K) m + n + 1. Lemma A.5. If d(K) 1, then e(K) 2d(K) - 1. Proof.We can assume K is a d = d(K) dimensional C.W. complex. Let L be its d - 1 skeleton, and let ac ac fi: Sd-1 ! L i=1 i=1 MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 11 denote the attaching maps of the d_cells of K. The complex L can be embedded in R2d-1, and we let U be a regular neighborhood. Thus L ,! U is an equivalence, and U is a 2d-1 dimensional parallelizable manifold. The composite ac ` c f Sd-1 --i=1-i--!L ,! U i=1 ` c is homotopic to an embedding, and then (U, i=1Sd-1) will represent K. Slava Krushkal has told us of an unpublished result of Stallings [S] that says that any d-dimensional and c-connected finite complex is (simple) ho- motopy equivalent to a subcomplex of R2d-c. This implies our final upper bound. Lemma A.6. e(K) 2d(K) - c(K). A.2. Lower bounds. The obvious lower bound for E(K) comes from di- mension: d(K) e(K). Stronger lower bounds arise from the contrapositive forms of the following proposition and corollary. Proposition A.7. If n e(K), then MapS (K, Sn) is equivalent to a sus- pension spectrum. Here MapS (K, X) denotes the function spectrum of stable maps between K and X, so that MapS (K, Sn) is the n-dual of K. Proof.It is easy to check (see e.g. [G1 ]) that the degree 1 approximation to 1 MapT (K, X) is MapS (K, X). Then, as in x3, we can conclude that, if n e(K) then the composite 1 F1C(K, S0) ! 1 C(K, S0) ! 1 MapT (K, Sn) ! MapS (K, Sn) is an equivalence, where F1C(K, S0) is the first filtration of the configuration space C(K, S0). The implications of the proposition for homology are the following. Corollary A.8. Suppose n e(K). (1) The reduced integral cohomology groups of K satisfy ( H~m (K; Z) is 0 ifm > n free abelian ifm = n. (2) For all primes p, Fp nH~-*(K; Fp) admits the structure of an unstable algebra over the mod p Steenrod algebra. 12 N.J.KUHN For certain two cell complexes, the lower bound of the proposition matches our upper0bounds.0Call a map f : Sm+k ! Sm stably minimal if whenever f0 : Sm +k ! Sm is a map so that f and f0 represent the same element in the stable homotopy group ßSk, one has m m0. Theorem A.9. If f : Sn ! Sm is stably minimal, and K is the mapping cone of f, then e( rK) = m + n + 1 + r. Proof.As S-duality induces the identity on ßS*, one deduces that MapS ( rK, Sr+m+n+1 ) ' MapS (K, Sm+n+1 ) ' 1 K. Since f is stably minimal, -i 1 K is not a suspension spectrum for any i > 0. The proposition thus implies that e( rK) m + n + 1 + r. The theorem follows, as Corollary A.4 and Corollary A.2 combine to show that e( rK) m + n + 1 + r. A.3. Examples. Theorem A.9 applies to all the classic 2-cell complexes. Explicitly, we have (1) e(Mr+2(d)) = e( rM2(d)) = 3 + r, (2) e( rCP 2) = 6 + r, (3) e( rHP 2) = 12 + r, (4) e( r(Cayley plane)) = 24 + r, and (5) e( r(D2p+1[ffS3)) = 2p+4+r if p is an odd prime and ff 2 ß2p(S3) is an element of order p. References [AK]S.T. Ahearn and N. J. Kuhn, Product and other fine structure in polynomial * *reso- lutions of mapping spaces, Alg. Geom. Topol. 2 (2002), 591-647. [Ar]G. Arone, A generalization of Snaith-type filtration , Trans. A. M. S. 351(* *1999), 1123-1250. [Bö]C.-F. Bödigheimer, Stable splitting of mapping spaces, Springer L. N. Math.* * 1286 (1987), 174-187. [B1]A. K. Bousfield, The localization of spaces with respect to homology, Topol* *ogy 14(1975), 133-150. [B2]A. K. Bousfield, On ~-rings and the K-theory of infinite loop spaces, K-The* *ory 10 (1996), no. 1, 1-30. [B3]A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math 119(199* *7), 1321- 1354. [EKMM] A.D.Elmendorf, I.Kriz, M.A.Mandell, J.P.May, Rings, modules, and algebr* *as in stable homotopy theory, A. M. S. Math. Surveys and Monographs 47, 1997. [G1]T. G. Goodwillie, Calculus I: the first derivative of pseudoisotopy, K-theo* *ry 4 (1990), 1-27. [G2]T. G. Goodwillie, Calculus II: analytic functors, K-theory 5 (1992), 295-33* *2. [G3]T. G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, Ge* *om. Topol. 7 (2003), 645-711. [K1]N. J. Kuhn, Suspension spectra and homology equivalences, Trans. A. M. S. 2* *83 (1984), 303-313. [K2]N. J. Kuhn, Localization of Andr'e-Quillen-Goodwillie towers, and the perio* *dic ho- mology of infinite loopspaces, preprint, 2003. MAPPING SPACES AND HOMOLOGY ISOMORPHISMS 13 [LS]L. Langsetmo and D. Stanley, Nondurable K-theory equivalence and Bousfield * *local- ization, K-theory 24 (2001), 397-410. [Ma]J. P. May, The geometry of interated loop spaces, Springer L. N. Math. 271,* * 1972. [McD] D. Mc Duff, Configuration spaces of positive and negative particles, Topo* *logy 14 (1975), 91-107. [S]J. R. Stallings, The embedding of homotopy types into manifolds, unpublished* * 1965 paper, available at http://math.berkeley.edu/~stall/. Department of Mathematics, University of Virginia, Charlottesville, VA 2290 E-mail address: njk4x@virginia.edu