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
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*omotopy Theory
and Their Applications III. Lecture Notes in Mathematics 99, Springer, Be*
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York 1969, p. 139-153
[BV] Boardman, J.M., Vogt, R.M. : Homotopy invariant algebraic structures on t*
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- Lecture Notes in Mathematics 347, Springer, Berlin Heidelberg New York *
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ON AN ADJOINT FUNCTOR TO THE THOM FUNCTOR 5
[C] Cohen, R. : The immersion conjecture for differentiable manifolds. - An*
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[H] Husemoller, D. : Fibre bundles. - McGraw-Hill, New York 1966
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*46
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[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
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