ON AN ADJOINT FUNCTOR TO THE THOM FUNCTOR Yuli B. Rudyak March 1999 Abstract. We construct a right adjoint functor to the Thom functor, i.e., * *to the functor which assigns the Thom space T to a vector bundle . Introduction Let o denote the functor which assigns the Thom space T to a vector bundle , and similarly for maps. The goal of this paper is to construct the right adjo* *int functor to the functor o. To motivate this result, I remark that it is always nice to know whether a f* *unctor admits an adjoint one. However, here we have more interesting motivation. Namel* *y, it is useful to know when a space is the Thom space of a certain vector bundle (spherical fibration). For example, de-thomification plays the important role * *in theory of immersion of manifolds, see [BP], [C] and a survey [L]. In fact, Brow* *n- Peterson [BP] make the de-thomification of a space, while Cohen [C] makes the de-thomification of a map. However, these de-thomifications are very ad hoc. So, it is reasonable to have a de-thomification machine, like the de-looping machin* *e of May [M] or Boardman-Vogt [BV]. The following observation of Beck [B] plays the crucial role in the de-looping theory. The suspension functor S is the left adj* *oint to the loop functor , and so there is a monad M := S. Clearly, every loop space is a space over M. Conversely, if a space X is a space over M then, using the simplicial resolution of the M-space X, one can provide a de-looping of X "* *at the simplicial level", and then certain additional arguments enable us to lift * *this "simplicial de-looping" to the geometric level, see [B], [M]. _____________ 1991 Mathematics Subject Classification. Primary 55R25, Secondary 18A40. 1 2 YULI B. RUDYAK Here we have a dual situation. As usual, the functor C := o is a comonad, and every Thom space is a space over C. Conversely, if X is a space over C then, du* *ally to what we said above, one can take the cosimplicial resolution X and provide a de-thomification of X "at the cosimplicial level". However, in order to do t* *he next step, a lifting to the geometrical level, one must prove that the thomific* *ation commutes with the functor Tot, and this problem looks quite complicated, cf. [B* *o]. Summarizing, one can consider this paper as a first step in approaching to t* *he attack of the de-thomification problem. Notice that the above arguments enable us to prove that a certain space is n* *ot a Thom space: it suffices to check that it is not a space over the comonad C. F* *or example, we have used (implicitly) these arguments in [R2] in order to prove th* *at the spectra k and kO are not Thom spectra. The case of non-orientable bundles Let On be the group of orthogonal transformations of the Euclidean space Rn, let BOn denote its classifying space, and let fl denote the universal n-dimensi* *onal vector bundle over BOn. Given a locally trivial bundle with the fiber Rn and structure group On, let T denote the Thom space of , i.e., T := D()=S() where D() is the total space of the unit disc bundle and S() is the total space* * of the unit sphere subbundle of D(). We regard T as a pointed space with the base point given by S(). Let K be the category whose objects are maps f : B ! BOn, where B is a connected space and f is a map such that ss1(B) -f*!ss1(BOn) = Z=2 is an epimorphism, and whose morphisms are commutative diagrams B ---'-! C ? ? f?y ?yg BOn ________BOn ; where f and g are objects of K. Let S be the category whose objects are pointed spaces X with ssi(X) = 0 for i < n and ssn(X) = Z=2 and whose morphisms are maps f : X ! Y such that f* : ssn(X) ! ssn(Y ) is an isomorphism. Let o : K ! S be the Thom functor which assigns the object of := T (f*fl) 2 S to the object f : X ! BOn of K. ON AN ADJOINT FUNCTOR TO THE THOM FUNCTOR 3 Theorem 1. The functor o admits a right adjoint functor : S ! K. Proof. We construct as follows. Choose any X 2 S. Given an integer k, let nkX be the component of nX corresponding to k 2 ssn(X) = ss0(nX) = Z=2. The standard On-action on Rn yields the obviousnOn-action on Sn which, in turn, induces a (right) On-action on nX = (X; *)(S ;*), and it is clear that ev* *ery component nkX; k = 0; 1 is On-invariant. Convert the right On-action on n1X into a left On-action by setting ga = ag-1 ; g 2 On; a 2 n1X. Consider the loca* *lly trivial bundle p : EOn xOn n1X ! BOn which is associated with the universal principal On-bundle := {EOn ! BOn}, cf. [PS]. We define X to be the map p. The -action on morphisms is clear. We prove that is right adjoint to o, i.e., that K(f; X) = S(T (f*fl); X) for every f : B ! BOn, cf. [R]. Indeed, consider the principal On-bundle f* = {q : E ! B}; and let be the n1X-bundle associated with f*, i.e. = {E xOn n1X ! B}: Then is induced by f from the bundle X = {p : EOn xOn n1X ! BOn}. So, K(f; X) = Sec where Sec denotes the set of all sections of . For every b 2 B choose any On-equivariant map ib : On ! E with qib(On) = b. We have (the first equality can be found e.g. in [H]) Sec ={On-equivariant maps E ! n1X} n ={On-equivariant maps f : E ! (X; *)(S ;*) such that f(x) 2 n1X for every a 2 E} ={maps f : E xOn (Sn ; *) ! (X; *) such that the map (Sn ; *) = On xOn (Sn ; *) -ib!E xOn (Sn ; *) -f!(X; *) belongs to n1X for every b} =S(T (f*fl); X). The case of orientable bundles Let BSOn be the classifying space for the connected component SOn of On. Let K0be the category whose objects are maps f : B ! BSOn, where B is a connected 4 YULI B. RUDYAK space, and whose morphisms are commutative diagrams B ---'-! C ? ? f?y ?yg BSOn ________BSOn ; where f and g are objects of K0. Let S0 be the category whose objects are pairs (X; aX ) where X is a pointed space with ssi(X) = 0 for i < n and aX is a gener* *ator (one of two) of ssn(X) = Z, and whose morphisms are maps ' : X ! Y with '*(aX ) = aY . Let fl0 be the universal oriented n-dimensional vector bundle over BSOn. The* *re is a unique element a 2 ssn(T fl0) = Z such that = 1 where u 2 Hn (T * *fl0) = Z is the orientation of fl0, h : ssn(T fl0) ! Hn(T fl0) is the Hurewicz homomor* *phism and <-; -> is the Kronecker pairing. Given an object f : X ! BSOn of K0, we have the canonical map F : T (f*fl0) ! T fl0, and F* : Z = ssn(T (f*(fl))) ! ssn(T fl) = Z is an isomorphism. Now defi* *ne the Thom functor o0 : K0! S0 by setting o0f = (T (f*fl); (F*)-1 (a)). Theorem 2. The functor o0 admits a right adjoint functor 0: S0 ! K0. Proof. Given an object (X; aX ) of K0, consider the isomorphism ssn(X) ~= ss0(nX), and let n1X be the component of nX which corresponds to aX . As in x1, we have the left SOn-action on nX, and it is clear the component n1X is invariant under the SOn-action on nX. We construct a fibre bundle p : ESOn xSOn n1X ! BSOn, and we define 0(X; aX ) := p. Now the proof can be completed similarly to 1.1. References [B] Beck, J, : On H-spaces and infinite loop spaces. - In: Category Theory, H* *omotopy Theory and Their Applications III. Lecture Notes in Mathematics 99, Springer, Be* *rlin Heidelberg New York 1969, p. 139-153 [BV] Boardman, J.M., Vogt, R.M. : Homotopy invariant algebraic structures on t* *opological spaces. - Lecture Notes in Mathematics 347, Springer, Berlin Heidelberg New York * *1973 [Bo] Bousfield, A.K. : On the homology spectral sequence of a cosimplicial spa* *ce. - Amer. J.Math. 109, 361-394 (1987) [BP] Brown, E.H., Peterson, F.P. : A universal space for normal bundles of n-m* *anifolds. - Comment. Math. Helv. 54, 405-430 (1979) ON AN ADJOINT FUNCTOR TO THE THOM FUNCTOR 5 [C] Cohen, R. : The immersion conjecture for differentiable manifolds. - An* *n of Math. 122, 237-328, (1985) [H] Husemoller, D. : Fibre bundles. - McGraw-Hill, New York 1966 [L] Lannes, J. : La conjecture des immersions (d'apres R.L. Cohen, E.H.Brown,* * F.P.Peterson et al. - In: Bourbaki Seminar 1981/1982, Soc. Math. France, Paris 1982, p. 331-3* *46 [M] May, J.P. : The geometry of iterated loop spaces. Lecture Notes in Mathem* *atics 271, Springer, Berlin Heidelberg New York 1972 [PS] Patterson, R., Stong, R. : Orientability of bundles. - Duke Math. J. 39, * *619-622 (1972) [R1] Rudyak, Yu.B. : On the Thom-Dold isomorphism for non-orientable vector bu* *ndles. - Soviet Math. Doklady 22, 3, 842-844 (1980) [R2] Rudyak, Yu.B. : The spectra k and kO are not Thom spectra. - In: Group* * Representa- tions:Cohomology, Group Actions and Topology, Proc. Symp. Pure Math. 63, * *Amer. Math. Soc, Providence, RI 1998, p. 475-483 Universit"at-GH Siegen, FB6/Mathematik, Emmy-Noeter Campus, Walter-Flex Str. 3, 57072 Siegen, Germany E-mail address: rudyak@mathematik.uni-siegen.de, july@mathi.uni-heidelberg.de