ADDENDUM TO “FINITE DECOMPOSITION
COMPLEXITY AND THE INTEGRAL NOVIKOV
CONJECTURE FOR HIGHER ALGEBRAIC K–THEORY”
DANIEL A. RAMRAS, ROMAIN TESSERA, AND GUOLIANG YU
Abstract. We supply an argument that was missing from the proof of
the main result of the article “Finite decomposition complexity and the
integral Novikov conjecture for higher algebraic K–theory” (J. Reine
Angew. Math., 694:129–178, 2014). The argument is essentially formal,
and does not affect the strategy of the proof.
We assume familiarity with the notation and definitions from Ramras–
Tessera–Yu [4]. The main technical result from [4] is the following vanishing
result for controlled K–theory.
Theorem ([4, Proposition 6.11]) If X is a bounded geometry metric space,
then Dγ (X) is vanishing for every ordinal γ.
The proof proceeds by transfinite induction on γ. It is stated in [4, p.
27] that “If γ is a limit ordinal and Proposition 6.11 holds for all β < γ,
it follows immediately from the definitions that Proposition 6.11 also holds
for γ.” It was pointed out by Daniel Kasprowski (private communication)
that this is an oversimplification: if Z ∈ Dγ (X), we can not immediately
conclude that Z ∈ Dβ for some β < γ. The notation Z ∈ Dγ (X) is highly
abusive; it means
only that each family {Zαr }α∈Ar is an element of the set
S
Dγ (X) := β<γ Dβ (X). Thus Z ∈ Dγ (X) means that for each r, there
exists βr < γ such that {Zαr }α∈Ar ∈ Dβr (X), but it is possible that the least
upper bound of the ordinals βr is actually γ. The goal of this addendum,
then, is to provide a complete discussion of the limit ordinal step in the proof
of Proposition 6.11. The argument presented here is due to Kasprowski, and
we thank him for sharing his ideas.
To avoid further abuse of notation, from here on we will replace the
notation Z ∈ Dη (X) by the statement “Z has complexity at most η.” Let γ
be a limit ordinal. Assume that for all β < γ, every decomposed sequence
W in X with complexity at most β is a vanishing sequence. Let Z be a
decomposed sequence in X with complexity at most γ. We must prove that
Z is in fact a vanishing sequence, in the sense that
(1)
colim K∗ Ac (Ps (Z)) = 0.
s∈Seq
1
2
RAMRAS, TESSERA, AND YU
Notation. Given a family of metric spaces {W
` α }α∈A and a number s > 0,
let Ps ({Wα }α ) denote the Rips complex Ps ( α Wα ).
For each s ∈ Seq, we have the Karoubi sequence
!
a
r
(2)
Ss ,→ Ac
(Psr ({Zα }α )) −→ Ac (Ps (Z))
r
defining Ac (Ps (Z)). Applying K–theory and passing to the colimit along
s ∈ Seq gives a long exact sequence (note that this is a filtered colimit, so
it preserves exactness). Hence to prove (1), it suffices to show that
For each ∗ ∈ Z, colim K∗ (Ss ) = 0
(3)
s∈Seq
and
!!
(4)
For each ∗ ∈ Z, colim K∗
s∈Seq
a
Ac
Psr ({Zαr }α )
= 0.
r
To prove (3), note that Ss is the colimit, over r < R, of
Y
Ac (Psr ({Zαr }α )) .
r<R
`
For each r, we know that {Zαr }α ∈ Dβr`for some βr < γ. The space α Zαr
decomposes over the family {Zαr }α , so α Zαr ∈ Dβr +1 . Since βr < γ and γ
is a limit ordinal, we have βr + 1 < γ, so our induction hypothesis implies
that
colim K∗ Ac (Psr ({Zαr }α )) = 0.
s∈Seq
The desired result now follows from the fact that K–theory commutes with
filtered colimits and with (finite) products.
Remark 1. Strictly speaking, our induction hypothesis states only that every
decomposed sequence of complexity at most β (with β < γ) is a vanishing
sequence. As explained in [4, Section 6] (see in particular Diagram 6.3), it
follows that for every space W ∈ Dβ (X) (β < γ) we have
colim K∗ (Ps W ) = 0.
s∈Seq
Now we turn to the proof of (4). For each s ∈ Seq, there is a natural
inclusion of categories
!
a
Y
Ac
Psr ({Zαr }α ) ,→
Ac (Psr ({Zαr }α )) ,
r
r
and these inclusions induce a functor
!
a
Y
r
j: colim Ac
Psr ({Zα }α ) ,→ colim Ac (Psr ({Zαr }α )) .
s∈Seq
r
s∈Seq
r
ADDENDUM: “DECOMPOSITION COMPLEXITY AND ALGEBRAIC K–THEORY” 3
We will now define a functor in the opposite direction,
!
Y
C: colim Ac (Psr ({Zαr }α )) −→ colim Ac
s∈Seq
a
s∈Seq
r
Psr ({Zαr }α ) ,
r
which will be inverse to j. On objects, this functor is simply induced by the
inclusions
!
a
Ac (Psr ({Zαr }α )) ,→ Ac
Psr ({Zαr }α ) ,
r
which are compatible as s increases. Given a morphism
(φr )r
(Mr )r −−−→ (Nr )r
in the category
Y
Ac (Psr ({Zαr }α )) ,
r
let Dr < ∞ be the propagation of φr , and let s0r = max(sr , Dr ). Applying
the functor
ηsr ,s0r : Ac (Psr ({Zαr }α )) −→ Ac Ps0r {Zαr }α ,
we see that ηsr ,s0r (φr ) now has propagation at most 3, so (ηsr ,s0r (φr ))r is a
morphism
ηs,s0 ((Mr )r ) −→ ηs,s0 ((Nr )r )
in the category
!
Ac
a
Psr ({Zαr }α )
.
r
Since ηs,s0 ((Mr )r ) represents C((Mr )r ) and ηs,s0 ((Nr )r ) represents C((Nr )r ),
we may define C((φr )r ) to be the morphism represented by (ηsr ,s0r (φr ))r .
It follows from the definitions that C is well-defined on morphisms and
functorial, and also that C and j are inverses. Hence (4) is equivalent to the
statement that
!
Y
(5)
For each ∗ ∈ Z, K∗ colim Ac (Psr ({Zαr }α )) = 0.
s∈Seq
r
Examining the definitions, one sees there is an isomorphism of categories
Y
Y
colim Ac (Psr ({Zαr }α )) ∼
colim Ac (Psr ({Zαr }α )) .
=
s∈Seq
r
r
s∈Seq
(Note that on the right, the colimit over s ∈ Seq may be replaced by a
colimit over s ∈ N, since Ac (Psr ({Zαr }α )) depends only on the r–term of
4
RAMRAS, TESSERA, AND YU
the sequence s.) This yields
!
colim
K∗
s∈Seq
Y
Ac (Psr ({Zαr }α ))
!
∼
= K∗
Y
r
r
colim Ac (Psr ({Zαr }α ))
s∈Seq
Y r
∼
K∗ colim (Ac (Psr ({Zα }α ))) ,
=
s∈Seq
r
where we have used Carlsson’s theorem that K–theory commutes with infinite products (for connective K–theory this is proven in [2]; the proof is
extended to
K–theory in [1]). As discussed above, for each
`non-connective
r
r we have α Zα ∈ Dβr +1 , and βr + 1 < γ, so each term in the product
Y
Y K∗ colim (Ac (Psr ({Zαr }α ))) ∼
colim K∗ (Ac (Psr ({Zαr }α )))
=
r
s∈Seq
r
s∈Seq
vanishes. This proves (5), and completes the proof of (1).
Comments regarding L–theory.
The argument above applies equally well to algebraic L–theory, if one
invokes the theorem of Carlsson and Pedersen [3] that L–theory commutes
with infinite products when coefficient category A satisfies Kr (A) = 0 for all
r << 0. In particular, this fills the missing step in the proof of [4, Theorem
7.9]. (We also take this opportunity to note an error in the statement of that
result; the stated hypothesis “L−∞ (A) = 0 for all ∗ < −r” should instead
read “Kr (A) = 0 for all ∗ < −r.”)1
It should be noted that our proof of the L–theoretic analog of the bounded
Borel conjecture (that is, the L–theoretic version of [4, Theorem 7.1]) requires the Carlsson–Pedersen result, so one must again assume that K∗ (A) =
0 for all ∗ << 0. Since Winges has shown that there are in fact additive
categories with involution for which L–theory does not commute with infinite products [5], it would be interesting to know if there is a way to prove
the L–theoretic bounded Borel conjecture for FDC metric spaces without
invoking Carlsson’s theorem.
References
[1] Gunnar Carlsson. Bounded K-theory and the assembly map in algebraic K-theory. In
Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), volume
227 of London Math. Soc. Lecture Note Ser., pages 5–127. Cambridge Univ. Press,
Cambridge, 1995.
[2] Gunnar Carlsson. On the algebraic K-theory of infinite product categories. K-Theory,
9(4):305–322, 1995.
[3] Gunnar Carlsson and Erik Kjær Pedersen. Controlled algebra and the Novikov conjectures for K- and L-theory. Topology, 34(3):731–758, 1995.
1We thank Christoph Winges for pointing out this misstatement.
ADDENDUM: “DECOMPOSITION COMPLEXITY AND ALGEBRAIC K–THEORY” 5
[4] Daniel A. Ramras, Romain Tessera, and Guoliang Yu. Finite decomposition complexity
and the integral Novikov conjecture for higher algebraic K–theory. To appear in J.
Reine Angew. Math. arXiv:1111.7022.
[5] Christoph Winges. A note on the L-theory of infinite product categories. Forum Math.,
25(4):665–676, 2013.
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA
E-mail address: [email protected]
Laboratoire de Mathmatiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France
E-mail address: [email protected]
Department of Mathematics, Texas A&M University, College Station, TX
77843, USA, and Shanghai Center for Mathematical Sciences, Shanghai, P. R.
China
E-mail address: [email protected]