On the spectrum bo ^ tmf
Scott M. Bailey
March 2, 2009
Abstract
M. Mahowald, in his work on bo-resolutions, constructed a bo-
module splitting of the spectrum bo^ bointo a wedge of summands
related to integral Brown-Gitler spectra. In this paper, a similar split-
ting of bo^ tmf is constructed. This splitting is then used to under-
stand the bo*-algebra structure of bo*tmf and allows for a description
of bo*tmf.
1 Introduction
All cohomology groups are assumed to have coefficients in F2 and all spectra
completed at the prime 2 unless stated otherwise. Let A denote the Steenrod
n
algebra, and A(n) the subalgebra generated by {Sq1, Sq2, . .,.Sq2 }. Con-
sider the Hopf algebra quotient A==A(n) = A A(n)F2. Here the right action
of A(n) on A is induced by the inclusion and the left action on F2 by the
augmentation. Algebraically, one can consider the subsequent surjections
A ! A==A(0) ! A==A(1) ! A==A(2) ! A==A(3) ! . . .
and ask whether each algebra can be realized as the cohomology of some
spectrum. The case n 3 requires the existance of a non-trivial map
n+1-1 0
S2 ! S which cannot occur due to Hopf invariant one. For n < 3,
however, it is now well-known that each algebra can indeed be realized by
the cohomology of some spectrum:
H*HF2 ! H*HZ ! H*bo ! H*tmf
1
There are maps realizing the above homomorphisms of cohomology groups
tmf ! bo ! HZ ! HF2
In particular, the spectrum tmf is at the top of a "tower" whose "lower floors"
have been well studied in the literature, culminating with Mahowald's [6]
understanding of the spectrum bo ^ bo and Carlsson's [1] description of the
cohomology operations [bo, bo]. More difficult questions arise: What is the
structure of tmf ^ tmf? What are the stable cohomology operations of tmf?
We would like to understand the spectrum bo^tmf for a variety of reasons.
First, it might serve as a nice intermediate step towards understanding the
spectrum tmf ^ tmf. Furthermore, determining its structure comes with an
added bonus of understanding operations [tmf , bo] which may provide some
insight into understanding the cohomology operations of tmf . Second, the
splitting of bo ^ tmf has been instrumental to the author in demonstrating
the splitting of the Tate spectrum of tmf into a wedge of suspensions of bo.
Let B1(j) denote the jthintegral Brown-Gitler spectrum, whose homology
will be described as a submodule of H*HZ. Such spectra have been studied
extensively in the literature (see [2], [5], [9], for example). In particular,
MahowaldW[6] demonstrated the splitting of bo-module spectra bo ^ bo '
4j W 8i+4j
j 0 bo^ B1(j). Let = 0 j i B1(j). The main theorem of this
paper is the following
Theorem 1.1. There is a homotopy equivalence of bo-module spectra
bo ^ ! bo^ tmf (1)
The splitting is analogous to that of bo ^ bo of Mahowald and even
MO<8> ^ bo of Davis [3]. Its proof, therefore, contains ideas and results
from both. Section 2 deals with demonstrating an isomorphism on the level
of homotopy groups, which first requires an understanding of the left A(1)-
module structure of H*tmf . In Section 3, we construct a map of bo-module
spectra realizing the isomorphism of homotopy groups. Section 4 uses this
splitting along with pairings of integral Brown-Gitler spectra to explicitly
determine the bo*-algebra structure of bo*tmf and also identifies the coho-
mology bo*tmf.
2
2 The algebraic splitting
The E2-term of the Adams spectral sequence converging to the homotopy
groups of bo ^ tmf is given by
Ext s,tA(H*(bo ^ tmf), F2)) sst-s(bo ^ tmf). (2)
The Ext -group appearing in the above spectral sequence can be simplified
via a change-of-rings isomorphism:
Exts,tA(1)(H*tmf , F2)) sst-s(bo ^ tmf). (3)
Therefore, it suffices to understand the left A(1)-module structure of H*tmf .
Computations and definitions simplify upon dualizing. Indeed, the dual
Steenrod algebra, A*, is the graded polynomial ring F2[,1, ,2, ,3, . .].with
|,i| = 2i- 1. An equivalent problem after dualizing is determining the right
A(1)-module structure of the subring H*tmf A*. The homology of tmf as
a right A-module is given by Rezk [8]
H*tmf ~= F2[i81, i42, i23, i4, . .].. (4)
The generators ii = O,i, where O : A* ! A* is the canonical antiautomor-
phism. Define a new weight on elements of A* by !(ii) = 2i-1 for i 1. For
a, b 2 A* define the weight on their product by !(ab) = !(a) + !(b). Let
Ntmfkdenote the F2-vector space inside H*tmf generated by all monomials of
weight k with Ntmf0= F2 generated by the identity.
Lemma 2.1. As right A(2)-modules,
M
H*tmf ~= Ntmf8i
i 0
Proof. Certainly, the two modules are isomorphic as F2-vector spaces. To
see there is an isomorphismPof right A(2)-modules, note that the right action
of the total square Sq = i 0Sqi on the generators of H*tmf is given by:
i81. Sq= i81+ 1;
i42. Sq= i42+ i81+ 1;
i23. Sq= i23+ i42+ i81+ 1;
Xn
i
in . Sq= i2n-i
i=0
3
for n > 3. Since !(1) = 0, modulo the identity the total square preserves
n-1 2n-1
the weight of the generators of H*tmf . Note that i21 Sq = 1, hence the
total square over A(2) cannot contain a 1 in the expansion for dimensional
__
reasons. |__|
Consider the homomorphism V : A* ! A* defined on generators by
(
1, i = 0, 1;
V (ii) =
ii-1, i 2.
Restricting V to the subring H*tmf A* clearly provides a surjection
Vtmf : H*tmf ! H*bo. Let Mbo(4i) denote the image of Ntmf8iunder the
homomorphism Vtmf. It is generated by all monomials with !(iI) 4i. The
following proposition is clear.
Proposition 2.2. As right A(2)-modules
Ntmf8i~= 8iMbo(4i). (5)
Proof. Due to the weight requirements, Vtmf is injective when restricted to
Ntmf8i. Indeed, the exponent of i1 in each monomial is uniquely determined
__
by the other exponents. |__|
Additionally, if we denote by Nbokthe F2-vector space inside H*bo gener-
ated by all elements of weight k with Nbo0= F2 generated by the identity, we
have a similar lemma:
Lemma 2.3. As right A(1)-modules,
Mn
Mbo(4i) ~= Nbo4j.
j=0
Further restricting V to the subring H*bo provides a surjection Vbo :
H*bo ! H*HZ. Let MHZ(2j) denote the image of Nbo4junder V . This sub-
module is generated by all monomials with !(iI) 2j. As in Proposition 2.2
we have the identification
Proposition 2.4. As right A(1)-modules,
Nbo4j~= 4jMHZ(2j). (6)
4
Goerss, Jones and Mahowald [5] identify the submodule MHZ(2j) H*HZ
as the homology of the jth integral Brown-Gitler spectrum:
Theorem 2.5 (Goerss, Jones, Mahowald [5]). For j 0, there is a spectrum
B1(j) and a map
g
B1(j) -! HZ
such that
(i) g* sends H*(B1(j)) isomorphically onto the span of monomials of weight
2j;
(ii)there are pairings
B1(m) ^ B1(n) ! B1(m + n)
whose homology homomorphism is compatible with the multiplication
in H*HZ.
Remark 2.1. The submodules Mbo(4i) are the so-called bo-Brown-Gitler
modules. There is a family of spectra with similar properties, having these
modules as their homology. Proposition 2.2 demonstrates that as an A(2)-
module, H*tmf is a direct sum of these modules. On the level of spectra,
however, tmf ^ tmf does not split as a wedge of bo-Brown-Gitler spectra.
Combining the results of Lemmas 2.1 and 2.3 with Theorem 2.5, H*tmf as
a right A(1)-module can be written in terms of homology of integral Brown-
Gitler spectra:
Theorem 2.6. As right A(1)-modules,
M
H*tmf ~= 8i+4jH*B1(j).
0 j i
The E2-term of the Adams spectral sequence (3) then becomes isomorphic
to M
8i+4jExts,tA(1)(H*B1(j), F2)) sst-s(bo ^ tmf). (7)
0 j i
This is precisely the Adams E2-term converging to the homotopy of bo ^ .
The chart can be obtained by applying the following theorem of Davis [4]
which links bo^ B1(n) to Adams covers of bo or bsp, depending on the parity
of n.
5
P s __
TheoremP2.7 (Davis [4]). If __n= (n1, . .,.ns), let |__n| = i=1ni and ff(n ) =
s __ V s
i=1ff(ni), and B1(n ) = i=1B1(ni). Then there are homotopy equivalences
( _ _
bo2|n|-ff(n), if |__n| is even;
bo ^ B1(__n) ' K _ _ _
bsp2|n|-1-ff(n),if |__n| is odd;
where K is a wedge of suspensions of HF2.
Figure 1: Ext s,tA(H*(bo ^ tmf), F2)) sst-s(bo ^ tmf)
The charts for bo and bsp are well known. Using the above theorem
along with the algebraic splitting of H*tmf , we see that Adams covers of bo
begin in stems congruent to 0 mod 8 while Adams covers of bsp begin in
stems congruent to 4 mod 8. The first 32 stems of the chart for bo ^ tmf is
displayed in Figure 1 modulo possible elements of order 2 in AdamsJfiltration
s = 0 corresponding to free A(1)'s inside H*tmf . The symbol appears
in Figure 1 to reduce clutter. It is used to mark the beginning of another
Z-tower. In general, all Z-towers are found in stems congruent to 0 mod 4
while those supporting multiplication by j occur in stems congruent to 4
mod 8.
Theorem 2.8. There is an isomorphism of homotopy groups
ss*(bo ^ tmf) ~=ss* (bo^ )
6
Proof. The E2-terms of their respective Adams spectral sequences have been
shown to be isomorphic. Both spectral sequences collapse. Indeed, the classes
charted in Figure 1 cannot support differentials for dimensional and natural-
ity reasons. Each element of order two in Adams filtration s = 0 correspond
to copies of A(1) inside H*tmf . These summands split off, obviating the
__
existance of differentials. |__|
3 The topological splitting
Theorem 1.1 concerns a bo-module splitting of the spectrum bo ^ tmf. The
following observation will aid us in studying bo-module maps.
Lemma 3.1. Let X and Y be spectra. Then
[bo ^ X, bo^ Y ]bo= [X, bo^ Y ]
Proof. Let ubo : S0 ! bo and mbo : bo ^ bo ! bo denote the unit and the
product map of bo, respectively. Given f : bo ^ X ! bo ^ Y and g : X !
bo^ Y , the equivalence is given by the composites
f 7! f O (u ^ 1)
g 7! (mbo^ 1) O (1 ^ g)
__
|__|
The spectra (bo, mbo, ubo) and (tmf , mtmf, utmf) are both unital E1 -ring
spectra [7]. This induces a unital E1 -ring structure (bo ^ tmf, m, u). This
structure will play an important role in the proof of the main theorem. We
begin by defining an increasing filtration of via:
`n 1`
n = 8i+4jB1(j) (8)
j=0i=j
Notationally, it will be convenient to let B(j) = 12jB1(j), so that the fil-
tration (8) can be rewritten as
`n `
n = 8iB(j). (9)
j=0i 0
7
The proof of Theorem 1.1 will proceed inductively on n. We will assume the
i-1
existance of a bo-module map %2i-1: bo^ 2 ! bo^ tmf which is a stable
A-isomorphism through a certain dimension. The inductive step will be then
i+1-1
to construct a bo-module map %2i+1-1: bo ^ 2 ! bo ^ tmf which is a
stable A-isomorphism through higher dimensions. To do this, we will employ
the pairings given in Theorem 2.5(ii). Define the map
gm,n : 8nB(m) ! bo^ tmf (10)
to be the restriction of %2i-1to the summand 8nB(m). Denote by gm = gm,0.
Lemma 3.2. Let m = 2i and 0 n < m. Suppose there are bo-module
maps fm : bo^ B1(m) ! bo^ tmf and fn : bo^ B1(n) ! bo^ tmf inducing
injections on homology. Then there is a bo-module map
fm+n : bo^ B1(m + n) ! bo^ tmf
inducing an injection on homology.
Proof. For all 0 n < m, Theorem 2.7 supplies equivalences of bo-module
spectra
bo^ B1(m) ^ B1(n) ' (bo^ B1(m + n)) _ K (11)
where K is a wedge of suspensions of HF2. There are no maps [HF2, bo^ tmf]
so that the composite m O (fm ^ fn) lifts as a bo-module map to the first
summand
fm+n : bo^ B1(m + n) ! bo^ tmf
__
hence is also an injection in homology. |__|
i-1
Corollary 3.3. Suppose there are bo-module maps %2i-1 : bo ^ 2 !
bo^ tmf and g2i : bo ^ B(2i) ! bo ^ tmf inducing injections on homology.
Then there is a bo-module map
i+1-1
%2i+1-1: bo^ 2 ! bo^ tmf
inducing an injection on homology groups.
Proof. For 0 m 2i - 1 and n 0, there are bo-module maps g2i+m,n
inducing an injection in homology. These maps are obtained by applying
Lemma 3.2 to g2iand the restriction of %2i-1to the summand
gm,n : bo^ 8nB(m) ! bo^ tmf
__
The map %2i+1-1is the wedge of these maps. |__|
8
The following observation will simplify our calculations inside the Adams
spectral sequence.
Lemma 3.4. Let X and Y be spectra. Suppose F : bo^ X ! bo^ Y is given
by the composite (mbo^ Y ) O (bo ^ f) for some map f : X ! bo ^ Y . Then
F*(rx) = rF*(x) if r 2 bo* and x 2 bo*X.
__
Proof. By construction, the composite F is a bo-module map. |__|
In particular, the bo-module map %2i+1-1constructed in Lemma 3.2 in-
duces a map in homotopy groups in Adams filtration s = 0. The above
lemma allows us to apply the bo*-module structure to extend the morphism
into positive Adams filtrations. To complete the inductive step it suffices to
construct a map g2i: B(2i) ! bo ^ tmf inducing an injection on homology.
Indeed, we can then apply Corollary 3.3 to extend %2i-1to a bo-module map
i+1-1
%2i+1-1: bo^ 2 ! bo^ tmf.
To construct g2i, we will use g2i-1supplied by the inductive hypothe-
sis. Once again we will attempt to use the pairing of integral Brown-Gitler
spectra:
B1(2i-1) ^ B1(2i-1) ! B1(2i) (12)
to construct a map bo ^ B1(2i) ! bo ^ tmf. Unfortunately, Lemma 3.2 will
not apply. Indeed, the above pairings (12) are not surjective in homology
since the element corresponding to ii+3 inside H*B1(2i) is indecomposible.
To handle this case, we turn to a lemma of Mahowald [6] made precise by
Davis [3]:
Lemma 3.5 (Davis [3]). If n is a power of 2, let Fn = 8n-5M2'^ B1(1).
j
There is a map Fn -! bo^B1(n)^B1(n) such that the cofibre of the composite
1^j mbo^1^1
ffi : bo^ Fn --! bo ^ bo^ B1(n) ^ B1(n) -----! bo ^ B1(n) ^ B1(n)
is equivalent modulo suspensions of HF2 to bo ^ B1(2n).
Define mi-1 : bo ^ B(2i-1) ^ B(2i-1) ! bo ^ tmf to be the bo-module
map induced by the composite m O (g2i-1^ g2i-1). With Lemma 3.5 in mind,
consider the diagram:
i+4-5 _ffi//_ i-1 i-1 ____//_ i
bo ^ 2 M2'^ B1(1) bo^ B(2 ) ^ B(2 ) bo^ B(2 )
k k
mi-1|| k k gkk
fflffl|uukk 2i
bo ^ tmf
9
It suffices to show the composite mi-1ffi is nulhomotopic, since then mi-1
would then extend to the desired map g2i. The following theorem is essen-
tially due to Davis [3, Prop. 2.8], however modified to our context.
Theorem 3.6 (Davis, [3]). Suppose g2i-1: bo^ B(2i-1) ! bo^ tmf induces
an injection on homology. Then
i-1 i-1
ss2i+4-4 bo^ B(2 ) ^ B(2 ) ~=Z(2) (13)
with generator ff2i+4-4whose image under (mi-1)] is divisible by 2.
Proof. Since bo ^ tmf has the structure of an E1 -ring spectrum, the map
mi-1 factors through the quadratic construction on B(2i-1), i.e., there is a
map j making the the following diagram commute:
bo ^4D2(B(2i-1))4
iiii
ijiiii |
iiiii |
ii fflffl|
bo ^ B(2i-1) ^ B(2i-1)____mi-1_//bo^ tmf
where
D2(B(2i-1)) = S1 n 2 (B(2i-1) ^ B(2i-1)).
Here the 2-action on S1 is the antipode and the action on the smash product
interchanges factors. Using this factorization, it suffices to show that the
induced map j] in homotopy sends the generator in dimension 2i+4 - 4 to
twice an element of the homotopy of the quadratic construction. This is
__
proved by Davis [3]. |__|
Proof that Theorem 3.6 implies Theorem 1.1. Let [xi] 2 ssi(bo ^ tmf) for i =
0, 8, 12 denote the classes in bidegree (i, 0) in the E2-term displayed in Fig-
ure 1. The class [x12] does not support action by j so that x12 extends to a
map B(1) ! bo^ tmf. Upon smashing with bo, we get maps
g0 : bo^ B(0)! bo^ tmf
g0,1: 8bo ^ B(0)! bo^ tmf
g1 : bo^ B(1)! bo^ tmf
inducing injections in homology. In particular, Lemma 3.2 extends these to a
bo-module map %1 : bo^ 1 ! bo^tmf which is also an injection on homology.
10
Lemma 3.4 extends this morphism to positive Adams filtrations. Figure 1
demonstrates that modulo possible order 2 elements on the zero line, this
map accounts for all homotopy classes through the 23-stem. Hence, it is a
stable A-equivalence in this range.
For the purpose of induction, assume the existance of a bo-module map
i-1
%2i-1 : bo ^ 2 ! bo ^ tmf inducing a stable A-equivalence through the
(12(2i)-1)-stem. In particular, there is a map g2i-1: bo^B(2i-1) ! bo^tmf
of bo-module spectra inducing an injection on homology groups. Define mi-1
and ffi as above. We will show mi-1ffi ' *.
oooO ____Oo
oooo oooo
ooo ooooo
O ____O____O____O_____________________________________*
*__________oo
__________________________________________________*
*_____________________________________________________________________________*
*_________________________
0 1 2 3 4
Figure 2: H*(M2'^ B1(1))
Figure 2 shows the cell diagram for H*(M2'^ B1(1)). Since there are no
elements of positive Adams filtration in stems congruent to {5, 6, 7} mod 8
in the Adams spectral sequence converging to ss*(bo ^ tmf), the composite
i+4-5
mi-1ffi restricts to a map 2 M2'! bo^ tmf. Consider the composite
i+4-5 a0 2i+4-5 mi-1ffi
S2 -! M2'^ B1(1) ---! bo ^ tmf
i+4-5
restricting mi-1ffi to the bottom cell of 2 M2'. There are no elements of
positive Adams filtration in stems congruent to 3 mod 8 so this restriction
extends to the top cell
i+4-4 a1 2i+4-5 mi-1ffi
S2 -! M2'^ B1(1) ---! bo ^ tmf.
Theorem 3.6 indicates that the class (mi-1)](ffia1) is divisible by 2. Hence,
__
this map is nulhomotopic. Applying Corollary 3.3 gives the result. |__|
4 The bo -homology of tmf
Both bo and tmf have the structure of E1 -ring spectra, so that the smash
product bo^ tmf also inherits such a structure. The splitting of bo^ tmf into
11
pieces involving integral Brown-Gitler spectra gives a nice description of its
structure as a ring spectrum. Indeed, the pairing of the B1(j) is compatible
with multiplication inside H*HZ of which H*tmf is a subring. In particular,
the pairings of the integral Brown-Gitler spectra induce the ring structure of
bo^tmf . The induced structure on homotopy groups is given by the following
theorem:
Theorem 4.1. There is an isomorphism of graded bo*-algebras
bo*[oe, bi, ~i| i 0]
ss*(bo ^ tmf) ~=__________________________2 F (14)
(~bi - 8bi+1, ~bi- 4~i, jbi)
where |oe| = 8, |bi| = 2i+4 - 4, |~i| = 2i+4 and F is a direct sum of F2 in
varying dimensions.
Proof. Theorem 2.7 gives homotopy equivalences
bo^ B(n) ^ B(2i) ! K _ (bo ^ B(n + 2i))
for all n < 2i. In particular, the induced pairings
ss*(bo ^ B(n)) ss*(bo ^ B(2i)) ! ss*(bo ^ B(n + 2i))
provide an isomorphism for all n < 2i, modulo possible order 2 elements in
Adams filtration zero corresponding to free A(1) inside H*tmf . Therefore,
the homotopy classes inside bo, 8bo and bo ^ B(2i) for i 0 generate the
homotopy of ss*(bo ^ tmf). Hence, it suffices to examine the pairings
bo ^ B(2i) ^ B(2i) ! bo^ B(2i+1).
Figure 3 depicts the E2-term of the Adams spectral sequence converging to
bo*B(1) along with its generators as a bo*-module. With these generators,
we can determine the decomposibles inside bo*B(2i). Indeed, Lemma 3.5
provides us with a fiber sequence
i+5-4
bo^ B(2i) ^ B(2i) ! bo^ B(2i+1) ! bo^ 2 M2'^ B1(1) (15)
inducing a long exact sequence of Ext -groups. Figure 4 shows how to use
(15) to form the E2-page of bo ^ B(2i+1). The arrows represent subsequent
differentials and the dotted lines non-trivial extensions. The classes in black
are those contributed by bo ^ B(2i) ^ B(2i), i.e., the decomposible classes
12
Figure 3: Ext s,tA(1)(H*B(1), F2)
hit by multiplication by elements in the summand bo ^ B(2i). Those in red
i+5-4
(or grey) are contributed by bo ^ 2 M2'^ B1(1). Denote by bi+1 the
class found in bidegree (2i+5- 4, 0) and ~i+1 the class in (2i+4, 1). These two
elements are thus indecomposible in the ring ss*(bo ^ tmf).
Note that the class in bidegree (2i+5- 8, 0) corresponds to the element
b2i. In particular, ~b2i= 8bi+1. Also note that ~bi+1 = 4~i+1 and jbi+1 = 0.
Figure 4: Ext s,tA(1)(H*B(2i+1), F2)
__
|__|
Remark 4.1. The splitting of bo^ tmf can also be used to give a description
of the bo-cohomology of tmf . Indeed, Lemma 3.1 gives that [tmf , bo] =
[bo ^ tmf, bo]bo. Since Theorem 1.1 provides a splitting as bo-module spectra,
13
one has the following chain of equivalences of bo*-comodules:
bo*tmf = [tmf , bo]
= [bo ^ tmf, bo]bo
" #
`
= 8nbo ^ B(m), bo
m,n 0 bo
" #
`
= 8nB(m), bo
m,n 0
M
= -8nbo*B(m)
m,n 0
A complete description of the summands bo*B(m) is given by Carlsson [1].
The comultiplication on bo*tmf is once again induced by the pairings of inte-
gral Brown-Gitler spectra. It would be interesting to determine the explicit
generators and relations as a bo*-coalgebra.
References
[1]Gunnar Carlsson, Operations in connective K-theory and associated co-
homology theories, Ph.D. thesis, Stanford, 1976.
[2]Fred R. Cohen, Donald M. Davis, Paul G. Goerss, and Mark E. Ma-
howald, Integral Brown-Gitler spectra, Proc. Amer. Math. Soc. 103
(1988), no. 4, 1299-1304.
[3]Donald M. Davis, The splitting of BO<8> ^ bo and MO<8> ^ bo, Trans.
Amer. Math. Soc. 276 (1983), no. 2, 671-683.
[4]Donald M. Davis, Sam Gitler, and Mark Mahowald, The stable geomet-
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113-132.
[6]Mark Mahowald, bo-resolutions, Pacific Journal of Mathematics 92
(1981), no. 2, 365-383.
14
[7]J.P. May, Infinite loop space theory, Bull. Amer. Math. Soc. 83 (1977),
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[8]Charles Rezk, Supplementary notes for Math 512 (ver. 0.18),
http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf, July 2007.
[9]Don H. Shimamoto, An integral version of the Brown-Gitler spectrum,
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15