THE WIRTHM ÜLLER ISOMORPHISM REVISITED
J.P. MAY
Abstract.We show how the formal Wirthmüller isomorphism theorem proven
in [2] simplifies the proof of the Wirthmüller isomorphism in equivarian*
*t stable
homotopy theory. Other examples from equivariant stable homotopy theory
show that the hypotheses of the formal Wirthmüller and formal Grothendie*
*ck
isomorphism theorems in [2] cannot be weakened.
Contents
1. The Wirthmüller isomorphism 2
2. The Adams isomorphism 4
3. Change of universe 5
4. Completion of the proof of the Wirthmüller isomorphism 5
References 9
We illustrate the force of the formal Wirthmüller isomorphism theorem of [2] *
*by
giving a worked example of an interesting theorem whose proof it simplifies, na*
*mely
the Wirthmüller isomorphism theorem in equivariant stable homotopy theory. It
relates categories of G-spectra and H-spectra for H G. We also say just a lit*
*tle
about the analogous Adams isomorphism that relates categories of G-spectra and
J-spectra for a quotient group J of G. That context gives an interesting situat*
*ion
where the formal hypotheses of the formal Wirthmüller isomorphism theorem hold
but the conclusion fails, showing that the more substantive hypothesis is essen*
*tial.
In their general form, the Wirthmüller and Adams isomorphisms are due to
Gaunce Lewis and myself [6]. It is a pleasure to thank Lewis for recent e-mail*
*s,
ongoing discussions, and his longstanding quest for simplifications and general*
*iza-
tions of these theorems. The analogy between these isomorphisms in topology and
Verdier duality was first explored by Po Hu [4], who carried out an idea of Lew*
*is
that these isomorphisms could be obtained using parametrized equivariant spectr*
*a.
She obtained a substantial generalization of the Wirthmüller isomorphism, but at
the price of greatly increased complexity. Exactly as in our proof here, the th*
*eory of
[2] allows considerable simplification of her work, and that is part of our mot*
*ivation.
The theory of [2] should also simplify the proof of the Wirthmüller isomorphism*
* in
A1 stable homotopy theory that Hu proved in [3]. Still another example recently
studied by Hu deals with change of universe. We show that it gives an interesti*
*ng
naturally occurring situation in which all but one of the hypotheses of the for*
*mal
Grothendieck isomorphism of [2] hold, but the conclusion fails because the rele*
*vant
left adjoint fails to preserve compact objects.
____________
The author was partially supported by the NSF.
1
2 J.P. MAY
1. The Wirthmüller isomorphism
Let H be a (closed) subgroup of a compact Lie group G and let f : H -! G
be the inclusion. Let L be the tangent H-representation at the coset eH 2 G=H.
Thus, if G is finite or, more generally, if H has finite index in G, then L = 0.
Let D and C be the stable homotopy categories of G-spectra and of H-spectra,
as constructed in [6] or, in more modern form, [7]. The category D depends on a
choice of a "G-universe" V on which to index G-spectra. We may think of V as a
chosen collection {V } of G-representations V that contains the trivial represe*
*nta-
tion and is closed under direct sums. The most interesting example is the compl*
*ete
G-universe obtained by allowing all representations of G. We think of represent*
*a-
tions as finite dimensional G-inner product spaces and let SV denote the one-po*
*int
compactification of V . The point of the choice of G-universe is that, in the c*
*on-
struction of D, we force suspension by some representations V to be equivalences
D -! D, and we must choose which representations to invert in this sense.
We insist that G=H embed in a representation V in our G-universe V , which
is otherwise unrestricted. We index H-spectra on the H-universe f*V = {f*V },
where f*V denotes the G-representation V viewed as an H-representation by pull-
back along f. Similarly, a G-spectrum Y gives an H-spectrum f*Y by pullback
along f. The functor f* has a left adjoint f!and a right adjoint f*. The former*
* is
usually written as either G+ ^H X or G nH X, and the latter is usually written *
*as
either FH (G+ , X) or FH [G, X). The Wirthmüller isomorphism reads as follows.
Theorem 1.1 (Wirthmüller isomorphism). There is a natural isomorphism
! : f*X -! f]X, where f]X = f!(X ^ S-L ).
That is, for an H-spectrum X, FH (G+ , X) ~=G+ ^H ( -L X).
Here the suspension H-spectrum of SL is invertible with inverse S-L , allowing
the definition -L X = X ^ S-L . Indeed, an embedding of G=H in V 2 V induces
an inclusion L* V of H-representions with orthogonal complement W . Since
SL ^ SW ~=Sf V is invertible in C , SL is also invertible in C .
The unit objects in C and D are the sphere spectra SH and SG . Both C and D
are closed symmetric monoidal categories under their smash product and function
spectrum functors ^ and F . It is immediate from the definitions that f* is str*
*ong
symmetric monoidal and commutes with F , as required in the Wirthmüller context
discussed in [2, xx2, 4].
Remark 1.2. In this context, the projection formula would assert that
Y ^ FH (G+ , X) ~=FH (G+ , f*Y ^ X),
which is false on the spectrum level but which holds on the stable category lev*
*el as
a consequence of the Wirthmüller isomorphism. Note that, on the spectrum level,
FH (G+ , X)G ~=XH , by a comparison of adjunctions.
The isomorphism [2, 4.3] required in the Wirthmüller context can be written
(1.3) D(G+ ^H SH ) ~=G+ ^H S-L .
It is a special case of equivariant Atiyah duality for smooth G-manifolds, whic*
*h is
proven by standard space level techniques (e.g. [6, IIIx5]) and is independent *
*of the
THE WIRTHMÜLLER ISOMORPHISM REVISITED 3
Wirthmüller isomorphism. By the tubular neighborhood theorem, we can extend
an embedding i : G=H -! V to an embedding
(1.4) ~i: G xH W -! V
of the normal bundle G xH W . Atiyah duality asserts that the G-space G=H+ is
Spanier-Whitehead V -dual to the Thom complex G+ ^H SW of the normal bundle
of the embedding. Desuspending by SV in D gives the required isomorphism (1.3).
The category D is triangulated, with distinguished triangles isomorphic to ca*
*non-
ical cofiber sequences of G-spectra. The triangulation is compatible with the c*
*losed
symmetric monoidal structure in the sense discussed in [8]. Moreover, D is com-
pactly generated. Writing Sn for the n-sphere G-spectrum, we can choose the
generators to be the G-spectra G=J+ ^ Sn, where J ranges over the (closed) sub-
groups of G and n runs over the integers. The same statements apply to C , and *
*the
functor f* is exact since it commutes with cofiber sequences on the level of sp*
*ectra.
Moreover, f* takes generators to compact objects. Indeed, this depends only on
the fact that the H-spaces G=J are compact and of the homotopy types of H-CW
complexes, although it is easier to verify using the stronger fact that the G=J*
* can
be decomposed as finite H-CW complexes.
Remark 1.5. Observe that the generators H=K+ of C and the generators G=J+ of
D, other than G=H+ , need not be dualizable if the universe is incomplete. In f*
*act,
Lewis [5, 7.1] has proven that G=J+ is dualizable if and only if G=J embeds in a
representation in V . Conceptually, it is compactness rather than dualizability*
* of
the generators that is relevant.
Digressively, there is an interesting conceptual point to be made about the c*
*hoice
of generators. There are (at least) three different, but Quillen equivalent, m*
*odel
categories of G-spectra. We can take "G-spectra" to mean G-spectra as originally
defined [6], SG -modules as defined in [1], or orthogonal G-spectra as defined *
*in
[7], where the cited Quillen equivalences are proven. These three categories a*
*re
also compactly generated in the model theoretic sense. In the first two cases, *
*the
generators in the model theoretic sense can be taken to be the generators in the
triangulated category sense that we have0just specified. As explained in [7], w*
*e can
alternatively take all G=J+ ^ SV -V for V, V 02 V as generators in the model th*
*e-
oretic sense. However, in the model category of orthogonal G-spectra, we not on*
*ly
can but must take this larger collection as generators in the model theoretic s*
*ense.
Nevertheless, the smaller collection suffices to generate the associated triang*
*ulated
homotopy category, since that is triangulated equivalent to the homotopy catego*
*ry
obtained from the other two model categories.
By the formal Wirthmüller isomorphism theorem [2, 6.1], to prove Theorem
1.1, it remains to prove that the map ! : f*X -! f]X specified in [2, 4.7] is an
isomorphism when X is a generating object H=K+ ^ Sn of C . Since it is obvious
that ! commutes with suspension and desuspension, we need only consider the case
n = 0, where the generators in question are the suspension H-spectra 1 H=K+ .
Here is the punchline. Suppose that G is finite or, more generally, that H has
finite index in G. Then, as an H-space, H=K+ is a retract of the G-space G=K+ .
The retraction sends cosets of G=K+ not in the image of H=K+ to the disjoint
basepoint of H=K+ . By [2, 4.13], it follows formally that ! is an isomorphism *
*when
X = 1 H=K+ . This simple argument already completes the proof of Theorem 1.1
in this case.
4 J.P. MAY
To prove Theorem 1.1 in general, we apply [2, 4.14], which allows us to work
one generating object at a time. This reduces all of our work to consideration *
*of
suspension spectra and thus of spaces. The argument is essentially the same as *
*part
of the argument in [6, IIIxx5, 6], but we shall run through the space-level det*
*ails in
x4 in order to have a readable and self-contained account.
2.The Adams isomorphism
Let N be a normal subgroup of a compact Lie group G and let f : G -! J,
J = G=N, be the quotient homomorphism. Fix a G-universe V = {V } and index
J-spectra on the N-fixed J-universe V N = {V N}. Regarding J-representations as
G-representations via f, we obtain a second G-universe f*V N = {f*V N}, and we
insist that f*V N be contained in the original G-universe V . Let D be the stab*
*le
homotopy category of J-spectra indexed on V N and let C be the stable homotopy
category of G-spectra indexed on f*V N.
Regarding J-spectra as G-spectra via f, we obtain a functor f* : D -! C . Its
left adjoint f!is just the orbit spectrum functor that sends a G-spectrum X ind*
*exed
on f*V N to X=N. Its right adjoint f* is just the fixed point spectrum functor *
*that
sends X to XN . The functor f* is strong symmetric monoidal, the isomorphism [2,
2.6] holds in the form (f*Y ^ X)=N ~=Y ^ (X=N), and f* takes generating objects
to compact objects. Since G-spectra in C are indexed on an N-trivial universe,
SG =N ~= SJ and [2, 4.3] holds in the trivial form D(SJ) ~= SG =N. Thus all of
the formal hypotheses of the formal Wirthmüller isomorphism theorem, [2, 6.1],
are satisfied. However, the conclusion fails, because ! is an isomorphism on so*
*me
but not all generators. Let C =N be the thick subcategory of C generated by all
n 1GG=H+ such that N H.
Proposition 2.1. For X 2 C =N, the natural map
! : XN - ! X=N
is an isomorphism of J-spectra.
Proof.The map ø : T -! f!C of [2, x4] is here just the isomorphism SJ ~=SG =N,
and the map ! of [2, 4.7] is just the composite of the isomorphism XN ~=f*XN =N
and "=N : f*XN =N -! X=N. When X = n 1GG=H+ with N H, the latter
map is also an isomorphism, but it is not an isomorphism in general.
The Adams isomorphism is more subtle. Let B be the stable homotopy category
of G-spectra indexed on V . We have the adjoint pair (i*, i*) of change of univ*
*erse
functors i* : C -! B and i* : B -! C induced by the inclusion i : f*V N -! V .
We are most interested in the composite adjunction (i*f*, f*i*). Thinking of the
case when V is a complete G-universe, it is usual to regard the composite f*i**
* :
B -! D as the N-fixed point spectrum functor from G-spectra indexed on V to
J-spectra indexed on V N.
The conjugation action of G on N gives rise to an action of G on the tangent
space A = A(N; G) of N at e. We call A the adjoint representation of G at N. Of
course, A = 0 if N is finite. The Adams isomorphism reads as follows. Let Cf be
the full subcategory of N-free G-spectra in C .
Theorem 2.2 (Adams isomorphism). For X 2 Cf, there is a natural isomorphism
! : f*i*i*X -! f]X, where f](X) = f!(X ^ SA ).
THE WIRTHMÜLLER ISOMORPHISM REVISITED 5
That is, for an N-free G-spectrum X indexed on f*V N, (i*i*X)N ~= A X=N.
It is usual to write this in the equivalent form (i* -A i*X)N ~= X=N, but the
present form is more convenient for applications and more sensible from the cat*
*e-
gorical point of view. This looks enough like the formal Wirthmüller isomorphism
to expect a similar proof. However, I do not have a helpful formal analysis.
3.Change of universe
Let i: V -! U be a map of G-universes, say for definiteness an inclusion.
We have the adjoint pair (i*, i*) relating the stable homotopy categories C and
D of G-spectra indexed on V and G-spectra indexed on U . The left adjoint i*
preserves compact objects, hence the right adjoint i* preserves coproducts [2, *
*5.4].
Since D is compactly generated, the right adjoint i* has a right adjoint i! by *
*the
triangulated adjoint functor theorem [2, 6.3]. It occurred to Po Hu to study the
structure of such functors i!, and the natural question to ask is whether or no*
*t the
formal Grothendieck isomorphism theorem [2, 6.4] applies. The functors (i*, i*,*
* i!)
here play the roles of the functors (f*, f*, f!) there, and the projection form*
*ula
takes the form of a natural isomorphism
Y ^ i*X ~=i*(i*Y ^ X).
This holds when Y is a suspension G-spectrum because the suspension G-spectrum
functors on the two universes satisfy i* 1 ~= 1 and i* commutes with smash
products. It therefore holds in general [2, 5.6].
There is a natural map
OE : i*Y ^ i!Z -! i!(Y ^ Z)
which is an isomorphism for all dualizable Y [2, 3.9]. We ask whether or not it*
* is an
isomorphism for all Y , and the answer is no. Indeed, the necessary hypothesis *
*that
i! preserves coproducts in [2, 6.4] is satisfied if and only if i* takes genera*
*tors to
compact objects [2, 5.4], and this fails in general. To see this, let U be a co*
*mplete G-
universe and V = U G be the G-fixed subuniverse {UG }, U 2 U . The G-fixed point
functor from D to the stable homotopy category of spectra is the composite of i*
** and
the G-fixed point spectrum functor from C to spectra. The latter functor preser*
*ves
compact objects by inspection. For example, it commutes with the suspension
spectrum functor and therefore takes suspension spectra of compact G-spaces to
suspension spectra of compact spaces. However, for a based G-space Y , the G-
spectrum 1 Y indexed on U has G-fixed point spectrum the wedge over conjugacy
classes (H) of the suspension spectra of the spaces EW H+ ^WH Ad(WH) Y H [6,
V.11.1]. Even when Y = S0, this spectrum is not compact. Therefore i* 1 Y
cannot be a compact G-spectrum indexed on U G.
4. Completion of the proof of the Wirthmüller isomorphism
We must verify the hypotheses of [2, 4.14] for X = 1 H=K+ . This means
that, with f]X = G+ ^H (X ^ S-L ), we must construct a map , : f*f]X -! X
such that certain diagrams commute. We need some space level constructions from
[6] to do this. The tubular neighborhood (1.4) gives a Pontryagin-Thom G-map
t : SV -! G+ ^H SW . It is V -dual to the counit G-map oe : f!f*S0 = G=H+ -! S0
[6, III.5.2]. The following construction, which is [6, II.5.5], specializes to *
*give the
6 J.P. MAY
V -dual u : G+ ^H SW -! SV to the unit H-map i : S0 -! f*f!S0 = G=H+ . We
omit f* from notations, instead stating the equivariance explicitly.
Construction 4.1. Let H x H act on G by (h1, h2)g = h1gh-12and act on L x H
by (h1, h2)(~, h) = (h1~, h1hh-12). We think of the first factor H as acting fr*
*om the
left, the second as acting from the right. Using the exponential map, construct*
* an
embedding j : L -! G of L as a slice at e such that
(4.2) j(h~) = hj(~)h-1 and j(-~) = j(~)-1.
Define ~j: L x H -! G by ~j(~, h) = j(~)h. Then ~jis an (H x H)-map that embeds
L x H onto an open neighborhood of e. Collapsing the complement to a point, we
obtain an (H x H)-map u : G+ -! SL ^ H+ . For a based H-space X, we obtain
an induced (left) H-map
u : G+ ^H X -! (SL ^ H+ ) ^H X ~=SL ^ X.
Setting X = SW and identifying SV with SL ^ SW , we obtain the promised V -dual
u : G+ ^H SW -! SV of i : S0 -! G=H+ .
We only need the following definition and lemmas for the H-spaces X = H=K+ ,
but it is simpler notationally to proceed more generally. We write suspension
coordinates on the right, V Y = Y ^ SV . This is important to remember for
control of signs, which are units in Burnside rings. Recall that, for a G-space*
* Y
and an H-space Z, the natural isomorphism of G-spaces
~ß: G+ ^H (Y ^ Z) -! Y ^ (G+ ^H Z)
is given by the formulas
(4.3) ~ß(g ^ y ^ z) = gy ^ g ^ z and ~ß-1(y ^ g ^ z) = g ^ g-1y ^ z.
Definition 4.4. For a based H-CW complex X, define an H-map
, : f*f] 1HX -! 1HX
as follows. Observe that we have natural isomorphisms of H-spectra
V f*f] 1HX ~=SV ^ (G+ ^H (X ^ S-L ))
~ß-1--!G V -L ~ 1 W
+ ^H (S ^ X ^ S ) = H (G+ ^H (X ^ S ))
and
V 1HX ~= 1H(X ^ SV ).
Let V , correspond under these isomorphisms to 1Hu, where
u : G+ ^H (X ^ SW ) -! SL ^ X ^ SW ~=X ^ SV .
The following observation is taken from [6, II.5.8].
Lemma 4.5. For a G-space Y and an H-space Z, the H-map
G+ ^H (Y ^ Z) ~=Y ^ (G+ ^H Z)id^u//_Y ^ SL ^ Z
is canonically H-homotopic to the H-map
G+ ^H (Y ^ Z) _u__//_SL ^ Y ^ Z ~=Y ^ SL ^ Z.
THE WIRTHMÜLLER ISOMORPHISM REVISITED 7
Therefore, for any H-map ` : Y - ! X from a G-space Y to an H-space X, the
following diagram is canonically homotopy commutative.
G+ ^H (Y ^ Z) ~= Y ^ (G+ ^H Z) id^u//_Y ^ SL ^ Z
id^(`^id)|| `^id||
fflffl| fflffl|
G+ ^H (X ^ Z) __u___//_SL ^ X ^ Z ~= X ^ SL ^ Z
Proof.Both maps send all points of G not in ~j(L x H) to the basepoint. The fir*
*st
takes ~j(~, h)^y^z to j(~)hy^~^hz and the second takes it to hy^~^hz. Applying
j to a contracting homotopy of L, we obtain an H-homotopy from j : L -! G to
the constant map at e, giving the required homotopy. The last statement follows
since the diagram commutes by naturality if its top arrow is replaced by u.
This leads to the following naturality statement. The partial naturality diag*
*ram
[2, 4.16] is the case in which Y = f*X and ` is the counit of the (f*, f*) adju*
*nction.
Lemma 4.6. Let Y be a G-spectrum and X be a based H-space. For any map
` : Y -! 1HX of H-spectra, the following diagram of H-spectra commutes in C .
,
G+ ^H (Y ^ S-L )_______//Y
id^(`^id)|| `||
fflffl| fflffl|
G+ ^H ( 1HX ^ S-L )__,__// 1HX
Proof.The upper map , in the diagram is defined formally in [2, 4.9]. It suffic*
*es
to prove that the diagram commutes after suspension by V . This has the effect *
*of
replacing S-L by SW on the left and suspending by V on the right. Comparing [2,
4.9] with Definition 4.4 and taking Z = SW , the conclusion reduces to applicat*
*ion of
the spacewise diagram of the previous lemma to the spaces that comprise the giv*
*en
spectra. Technically, this is most easily seen using prespectra or orthogonal s*
*pectra
rather than actual spectra, but the essential point is just that the homotopy i*
*n the
previous lemma is sufficiently natural to commute with the structure maps.
To complete the proof of Theorem 1.1, it suffices to show that the following
specialization of the diagram [2, 4.17] commutes.
(4.7) 1HX ^ S-L ______i______//f*f!( 1HXO^OS-L )
i|| |f*f!(,^id)|
fflffl| |
f*f!( 1HX ^ S-L )f*ø//_f*f!(f*f!( 1HX ^ S-L ) ^ S-L )
Here f!(-) = G+ ^H (-), f* is the forgetful functor, i is the unit of the (f!, *
*f*)
adjunction, and ø is the map defined in [2, 4.6] with Y = f!( 1HX ^S-L ). We sh*
*all
see that this reduces to the following space level observations from [6, IIx5].
Lemma 4.8. The following composite is H-homotopic to the identity map.
SV __t_//_G+ ^H SW_u__//SV .
8 J.P. MAY
Proof.Composing the embeddings ~i: G xH W -! V and ~j: L x H -! G, we
obtain an embedding V = L x W = (L x H) xH W -! G xH V . The composite
uOt is k-1 on k(V ), and it collapses the complement of k(V ) to the basepoint.*
* The
embedding k is isotopic to the identity, and application of the Pontryagin-Thom
construction to an isotopy gives a homotopy id' u O t.
Lemma 4.9. For an H-space X, the following diagram is H-homotopy commuta-
tive. Here oe : SL -! SL maps ~ to -~.
Vi //V V oo~ß-1 V
VX _________ (G+ ^H X) ~=S ^ (G+ ^H X)______G+ ^H (S ^ X)
| OO
| |~=
| |
| |
Vi|| G+ ^H WO(SLO^ X)
| | W
| |id^ (oe^id)u
fflffl| |
V(G+ ^H X) ___id^t_//(G+ ^H X) ^ (G+ ^H SW_)~ß-1_//G+ ^H W (G+ ^H X)
Proof.The composite around the bottom maps all points with V coordinate not in
~i(j(L)xW )) to the basepoint. It maps the point x^~i(j(~), w) to (j(~, x)^j(~)*
*(~, w);
the sign map oe enters due to (4.2) and (4.3). The H-contractibility of L impli*
*es
that an H-homotopic map is obtained if we replace f(~) by the identity element
e 2 G. Thus the composite around the right is H-homotopic to i ^ u O t, which is
H-homotopic to the identify by Lemma 4.8.
Proof of Theorem 1.1.We must show that the diagram (4.7) commutes. It suffices
to prove this after suspending by V and replacing X by V X. This has the effec*
*t of
replacing the H-spectra S-L that appear in the diagram by the H-spaces SW . Sin*
*ce
the functors appearing in the diagram commute with the respective suspension
spectrum functors, this reduces the problem to the space level. Here a slightly
finicky diagram chase, which amounts to a check of signs coming from permutatio*
*ns
of suspension coordinates, shows that the resulting diagram commutes by Lemma
4.9. One point is that oe ^ id: SL ^ SL -! SL ^ SL is H-homotopic to the
transposition via the homotopy given by multiplying by the (2 x 2)-matrices with
rows (-t, 1 - t) and (1 - t, t) for t 2 I. Another is that we must apply Lemma *
*4.9
with X replaced by X ^ SW , which has the effect of introducing a permutation of
SL past SW . In more detail, after applying 1H and replacing X by X ^ SW , the
bottom composite in the diagram of the previous lemma agrees with
V ø : V (G+ ^H ( V X ^ S-L )) -! V (G+ ^H (G+ ^H ( V X ^ S-L ) ^ S-L )
under the evident isomorphisms of its domain and target. The interpretation of *
*the
right vertical composite is trickier, because of the specification of
, : G+ ^H ( V X ^ S-L ) -! V X
in Definition 4.4. A diagram chase after suspending by V shows that , agrees un*
*der
the evident isomorphism of its source with
(oe ^ id) O u : G+ ^H (X ^ SW ) -! SL ^ X ^ SW ~= V X.
Notice that we have evident isomorphisms
1H V (G+ ^H (X ^ SW )) ~= V (G+ ^H ( V X ^ S-L )) ~= 1HG+ ^H ( V X ^ SW ),
THE WIRTHMÜLLER ISOMORPHISM REVISITED 9
the first of which internally expands SW to SV ^ S-L and the second of which
leaves V X alone but uses ~ßto bring SV inside and contracts S-L ^ SV to SW .
Their composite enters into the upper right corner of the required diagram chas*
*e;
the transpositions that appear cancel out others, resulting in a sign free conc*
*lusion.
With these indications, the rest is routine.
Remark 4.10. Effectively, this proof uses the space level arguments of [6, IIx5*
*],
but eliminates the need for the spectrum level arguments of [6, IIx6]. We warn
the reader that [6, II.5.2] and hence the first diagram of [6, II.6.12] are inc*
*orrect.
Fortunately, they are also irrelevant.
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