ON FIBRATIONS RELATED TO REAL SPECTRA
NITU KITCHLOO AND W. STEPHEN WILSON
Abstract.We consider real spectra, collections of Z=(2)-spaces indexed o*
*ver
Z Zff with compatibility conditions. We produce fibrations connecting *
*the
homotopy fixed points and the spaces in these spectra. We also evaluate
the map which is the analogue of the forgetful functor from complex to r*
*eals
composed with complexification. Our first fibration is used to connect t*
*he real
2n+2(2n-1)-periodic Johnson-Wilson spectrum ER(n) to the usual 2(2n-1)-
periodic Johnson-Wilson spectrum, E(n). Our main result is the fibration
~(n)ER(n) ! ER(n) ! E(n), where ~(n) = 22n+1- 2n+2+ 1.
1.Introduction
In 1968, Landweber introduced the idea of a real complex cobordism by tak-
ing the homotopy fixed points of complex cobordism under complex conjugation,
[Lan68]. A few years later this theory was studied again by Araki in [AM78 ],
[Ara79a], and [Ara79b]. Recently there has been a flurry of activity around th*
*is
theory by Hu and Kriz; [HK01a ], [HK02 ], [HK01b ], [HK ], [Hu01 ], and [Hu ].
Hu and Kriz (in [HK01a ]) produce real versions, ER(n), of the Johnson-Wilson
spectra E(n) ([JW73 ]) and compute their homotopy. The homotopy of E(n) is
Z(2)[v1, v2, . .,.vn1]. E(n) is periodic of period |vn| = 2(2n - 1), ER(n) is p*
*eriodic
of period |v2n+1n| = 2n+2(2n - 1), and the construction gives maps of spectra:
ER(n) ! E(n).
In the case n = 1 this is just the map KO(2)! KU(2)and Wood identified the
fibre as KO(2).
The main purpose of this paper is to identify the fibre of ER(n) ! E(n), pro-
ducing the fibration:
(1.1) ~(n)ER(n) ! ER(n) ! E(n).
where ~(n) = 22n+1- 2n+2 + 1. These fibrations should make these theories much
more accessible.
Let E be a real spectrum as defined in [HK01a ]. In particular, E is given by*
* a col-
lection of pointed Z=(2)-spaces EV indexed by the representation ring RO(Z=(2))
of the group Z=(2). Recall that RO(Z=(2)) = Z Zff, where ff is the sign repre-
sentation. Moreover, we require that the spaces EV be compatible in the followi*
*ng
sense:
Given a representation U, and a pointed Z=(2)-space X, let U X denote the
space Map *(SU , X), where SU is the one-point compactification of U. The space
U X has an induced diagonal action of the group Z=(2). For the spectrum E, we
require the existence of a family of equivariant homeomorphisms ffU,V : U EU V*
* !
EV , that satisfy obvious compatibility.
A multiplicative real spectrum E is one that admits a multiplication preservi*
*ng
the real structure (see [HK01a ]).
1
2 NITU KITCHLOO AND W. STEPHEN WILSON
Example 1.2. The real complex bordism spectrum MU is defined as follows. Let
MU(n) denote the Thom space of the universal bundle over BU(n). Complex
conjugation induces an action of Z=(2) on MU(n). Define MUV as the space
lim-!n n(1+ff)-VMU(n) for V 2 RO(Z=(2)). Notice that n(1 + ff) - V is a well
defined representation of Z=(2) for sufficiently large values of n. It is left*
* to the
reader to verify that MU has the properties of a multiplicative real spectrum.
Example 1.3. The Brown-Peterson spectrum has a real analogue BP. The real
Johnson-Wilson spectra E(n) may also be defined along similar lines [HK01a ].
These spectra are in fact multiplicative real spectra. E(1) is 2-localized rea*
*l K-
theory of Atiyah [HK01a ].
We will use the notation ERV to denote the homotopy fixed points of the Z=(2)-
action on EV . Notice that for a fixed V 2 RO(Z=(2)), the collection of spaces
{ERn+V , n 2 Z} form a spectrum in the usual sense. We shall abuse notation and
refer to the spectra {ERn+V , n 2 Z} and {En+V , n 2 Z} as the spectra ERV and
EV respectively. The purpose of this paper is to relate ERV to EV via a fibrati*
*on.
Of particular interest to us will be the case when the spectrum E is E(n). We h*
*ave:
Theorem 1.4. There are fibrations of spectra:
ERV -ffa-!ERV -'! EV , EV 1+oe-!ERV -a! ERV +ff
where the map a is induced by the map a : S0 -! Sffgiven by the inclusion of the
poles. The map ' is the standard inclusion, and the map (1+oe) is a lift of the*
* Norm
map on EV . Moreover, if E is a multiplicative real spectrum, then ERV is a ER0-
module spectrum for all V , and the above fibrations are fibrations of ER0-modu*
*le
spectra.
Remark 1.5. On the level of individual spaces we have fibrations ERm+(n-1)ff!
ERm+nff! Em+nff. This is a great help to computations as we hope to demonstrate
in a future paper.
Observe that the spaces EV -1= EV and EV -ff= ffEV are homeomorphic.
(Actually, Em+nff and Em0+n0ffare the same when m + n = m0+ n0.) In the
statement of the next theorem, we will use this homeomorphism to identify the t*
*wo
spaces. Note, however that the action of Z=(2) on the two spaces is different. *
*If
we let oe denote the action of the generator of Z=(2) on EV -1, and ~oethe acti*
*on on
EV -ff, then the two actions are related via ~oe= oeff* = -oe.
Now consider the boundary map. This map @ is defined as the map EV -1!
ERV -ffgiven by looping back the first fibration above composed with the map
ERV -ff! EV -ffgiven by the inclusion of the fixed points. Therefore
@ : EV -1-! EV -ff.
We have
Theorem 1.6. Let EV -1be identified with the space EV -ffas explained above.
Then the map @ is given by @ = Id - oe = Id + ~oe.
The standard example of this result is the composition KU ! KO ! KU and
this is just a generalization of it. The boundary is the composition of two map*
*s.
The first can be thought of as forgetting the complex structure and looking onl*
*y at
the underlying real structure. The next map can be thought of as complexificati*
*on.
ON FIBRATIONS RELATED TO REAL SPECTRA 3
This boundary map comes in useful in calculations we hope will appear in a futu*
*re
paper.
Our primary interest is the case when E = E(n). The following theorem uses
the computation of the homotopy of ER(n) given in [HK01a ].
Theorem 1.7. There exist nontrivial elements x(n) 2 ß~(n)(ER(n)0), where ~(n)
is the integer defined by ~(n) = 22n+1- 2n+2 + 1, such that one has a fibration*
* of
ER(n)0-module spectra:
~(n)ER(n)V x(n)-!ER(n)V -'! E(n)V .
Remark 1.8. An interesting special case of the above theorem is when n = 1, and
V = 0. Note that E(1) = KU(2), and hence ER(1) = KO(2). Moreover, the element
x(1) is none other than j. Hence one reproduces a well-known result
KO(2)-j!KO(2)-! KU(2).
More generally, fixing V = 0, we get the fibration (1.1).
Our dependence on the work of Hu and Kriz is obvious. In addition, we thank
them for numerous conversations.
2.The Fibrations
In this section we will show the existence of the two fibration given in the *
*intro-
duction.
Let Sffdenote the one-point compactification of the one dimensional nontrivial
representation of Z=(2). Notice that one has a Z=(2)-equivariant cofibration:
(2.1) Z=(2)+ -! S0 -a!Sff
where the map Z=(2)+ ! S0 is given by the pinch map. Let E be a real spectrum,
and for the purposes of this section, let EV denote the spectrum given by the
collection of spaces {En+V , n 2 Z}. Smashing the cofibration 2.1 yields a cofi*
*bration
of equivariant spectra
(2.2) EV ^ Z=(2)+ -! EV -a! EV +ff.
Notice that Z=(2)+ may be identified with S0 _ S0, with the Z=(2) action given *
*by
the twist map. Under this identification, the pinch map Z=(2)+ ! S0 corresponds
to the fold map S0 _ S0 ! S0. Hence, in the category of spectra, EV ^ Z=(2)+ may
be identified with EV _ EV = EV x EV with the Z=(2) action given by ~oe(x, y) =
(oe(y), oe(x)), where oe denotes the generator of Z=(2). Furthermore, the pinch*
* map
EV ^ Z=(2)+ ! EV corresponds to the sum map EV x EV -+! EV .
Consider the twisted diagonal map
: EV -! EV x EV , (x) = (x, oe(x)).
Notice that ~oe (x) = (x). From this it follows easily that lifts to an equi*
*valence
EV ! (EV x EV )hZ=(2). Putting these results together, we get
Proof of the second fibration in TheoremT1.4.aking homotopy fixed points of 2.2
yields another fibration. If we identify (EV ^ Z=(2)+ )hZ=(2)with EV , then the*
* map
(EV ^ Z=(2)+ )hZ=(2)! (EV )hZ=(2)is a lift of (1 + oe).
4 NITU KITCHLOO AND W. STEPHEN WILSON
Proof of the first fibration in TheoremF1.4.or the second fibration, one consid*
*ers
the Spanier-Whitehead dual of 2.1:
S-ff-a! S0 -! Z=(2)+
where the map S0 ! Z=(2)+ = S0 _ S0 corresponds to the diagonal. Smashing
with EV yields an equivariant fibration
EV -ffa-!EV -! EV x EV .
Taking homotopy fixed points of this fibration and making the identifications d*
*e-
scribed earlier, we get the remaining fibration:
ERV -ffa-!ERV -'! EV .
To complete the proof one simply observes that all the above constructions resp*
*ect
the ER0-module structure if E is a multiplicative real spectrum.
3.The Boundary map
In this section, we analyse the boundary map for the above fibrations. This m*
*ap
@ is defined as the composite of the map EV -1! ERV -ffgiven by looping back
the fibration constructed in the previous section, and the map ERV -ff! EV -ff
given by the inclusion of the fixed points. Therefore
@ : EV -1-! EV -ff.
The map @ may be explicitly constructed as follows. Consider the composite equi-
variant map given by the fold map followed by the pinch map:
(3.1) Z=(2)+ ^ Sff-f! Sff-p! Z=(2)+ ^ S1.
Notice that the Spanier-Whitehead dual of the pinch map p : Sff! Z=(2)+ ^ S1
is the difference map (-) : Z=(2)+ ^ S-1 ! S-ff. Taking the Spanier-Whitehead
dual of the composite (3.1) yields
Z=(2)+ ^ S-1 (-)-!S-ff-! Z=(2)+ ^ S-ff.
On smashing the above with EV , we obtain the composite map
(-) : EV -1x EV -1(-)-!EV -ff-! EV -ffx EV -ff.
From the previous section, we can see that there is a commutative diagram
EV -1 @ EV -ff
(1,oe) (1,~oe)
(-)
EV -1x EV -1 EV -ffx EV -ff
where oe denotes the Z=(2)-action on EV -1, and ~oedenotes the Z=(2)-action on
EV -ff. Recall that the spaces EV -1= EV and EV -ff= ffEV are homeomorphic
and the above two actions are related via ~oe= oeff*. Since ff* is homotopic to*
* the
inversion, we have ~oe= -oe. From a diagram chase we get @(x) = x-oe(x) = x+~oe*
*(x).
ON FIBRATIONS RELATED TO REAL SPECTRA 5
4. The case of E(n)
We recall the computation (via the Borel spectral sequence) of the homotopy
of BPR given in [HK01a ], and described in the form we need in [Hu01 ]. We will
reproduce the Borel specral sequence with BP replaced by E(n). The E2-term of
the Borel spectral sequence for E(n) is given by
E2 = Z(2)[vk, vn1, a, oe 2]=(2a), n > k 0, v0 = 2.
The bidegrees of the generators is given by
|a| = -ff, |vk| = (2k - 1)(1 + ff), |oe2| = 2(ff - 1).
The differentials are given by comparing with the Borel spectral sequence conve*
*rging
to the homotopy of BPR. In particular, the elements vk and a are permanent cycl*
*es,
and the nontrivial differentials are
k 2k+1-1
d2k+1-1(oe-2 ) = vka , 0 < k n.
Using the methods of [HK01a ], [Hu01 ], we notice that the E1 -term for the hom*
*o-
topy of ER(n) is given by the following ring:
k+1 1 2n+1
Z(2)[vkoel2 , a, vn , oe ]=I, n > k 0, l 2 Z
where I is the ideal generated by the relations:
v0 = 2,
k+1-1 l2k+1
a2 vkoe = 0,
m+1 s2m-k2k+1 (l+s)2m+1
vm oel2 .vkoe = vk.vm oe m k.
The bidegrees of the generators are given by
k+1 k k+1
|a| = -ff, |vkoel2 | = (2 - 1)(1 + ff) + l2 (ff - 1).
Comparing with the homotopy of BPR, we notice that there are no extension prob-
lems, and so the above is in fact isomorphic to the homotopy of ER(n).
Now consider the element
n-1 -2n+1(2n-1-1)
y(n) = v2n oe , y(n) 2 ß~(n)(ER(n)-ff)
where ~(n) is the integer ~(n) = 22n+1 - 2n+2 + 1. The element y(n) is clearly
invertible in the above ring. Hence we get
Claim 4.1. Multiplication by the element y(n) yields an equivalence of ER0-modu*
*le
spectra:
~(n)ER(n)V y(n)-!ER(n)V -ff.
We define the element x(n) to be the element
x(n) = a.y(n), x(n) 2 ß~(n)(ER(n)0).
This claim, along with the first fibration given in Theorem 1.4 yields the pr*
*oof
of Theorem 1.7.
Remark 4.2. The spectrum ER(n)0 is periodic with period 2n+2(2n - 1) generated
by the homotopy element v2n+1noe-2n+1(2n-1).
6 NITU KITCHLOO AND W. STEPHEN WILSON
References
[AM78] S. Araki and M. Murayama, fi-cohomology theories, Japan J. Math. (N.S.) *
*4 (1978),
no. 2, 363-416.
[Ara79a]S. Araki, Forgetful spectral sequences, Osaka J. Math. 16 (1979), no. 1*
*, 173-199.
[Ara79b]____, Orientations in fi-cohomology theories, Japan J. Math. (N.S.) 5 (*
*1979), no. 2,
403-430.
[HK] P. Hu and I. Kriz, Real cobordism and Greek letteer elements in the geom*
*etric chromatic
spectral sequence, preprint.
[HK01a]_____, Real-oriented homotopy theory and an analogue of the Adams-Noviko*
*v spectral
sequence, Topology 40 (2001), no. 2, 317-399.
[HK01b]_____, Some remarks on Real and algebraic cobordism, K-Theory 22 (2001),*
* no. 4,
335-366.
[HK02] _____, The homology of BPO , Recent Progress in Homotopy Theory: Proceed*
*ings
of a conference on Recent Progress in Homotopy Theory March 17-27, 2000,*
* Johns
Hopkins University, Baltimore, MD. (Providence, Rhode Island) (D. Davis,*
* J. Morava,
G. Nishida, W. S. Wilson, and N. Yagita, eds.), Contemporary Mathematics*
*, vol. 293,
American Mathematical Society, 2002, pp. 111-123.
[Hu] P. Hu, On Real-oriented Johnson-Wilson cohomology, preprint.
[Hu01] _____, The Ext0-term of the Real-oriented Adams-Novikov spectral sequenc*
*e, Homo-
topy Methods in algebraic topology (Providence, Rhode Island), Contempor*
*ary Mathe-
matics, vol. 271, American Mathematical Society, 2001, pp. 141-153.
[JW73] D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson *
*homology,
Topology 12 (1973), 327-353.
[Lan68]P. S. Landweber, Conjugations on complex manifolds and equivariant homot*
*opy of MU ,
Bulletin of the American Mathematical Society 74 (1968), 271-274.
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: nitu@math.jhu.edu, wsw@math.jhu.edu