Homological stability for moduli spaces of high dimensional manifolds. II

0.87 MB·

by  Soren Galatius

HOMOLOGICAL STABILITY FOR MODULI SPACES OF HIGH
DIMENSIONAL MANIFOLDS. II
SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
Abstract. We prove a homological stability theorem for moduli spaces of
manifolds of dimension 2n, for attaching handles of index at least n, after
thesemanifoldshavebeenstabilisedbycountablymanycopiesofSn×Sn.
Combined with previous work of the authors, we obtain an analogue of
6
the Madsen–Weiss theorem for any simply-connected manifold of dimension
1
2n≥6.
0
2

n
a
J 1. Introduction and statement of results

2
Let W be a smooth closed manifold of dimension 2n, and BDiff(W) denote

] the classifying space of the group of diffeomorphisms of W, equipped with the
T
Whitney C∞-topology. Our goal is to compute the cohomology of this space, and
A
one consequence of the results of this paper will be a map
.
h (1.1) α :BDiff(W)−→(Ω∞MTθ)//hAut(θ)
t W
a
m whichisanisomorphismonhomologyinarangeofdegrees,aslongasW issimply-
connected and 2n ≥ 6. The range of degrees in which it is an isomorphism will
[
increase with the genus

1
v g(W)=max{g ∈N | (cid:93)gSn×Sn is a connect summand of W},
2
and the right hand side of (1.1) is homotopy-theoretic and constructed purely in
3
2 terms of the Moore–Postnikov n-stage factorisation
0
0 W −(cid:96)W→B −θ→BO(2n)
.
1 for a Gauss map of W. It is well-suited to calculation.
0
This result will be obtained by combining our earlier work [GRW14b, GRW14a]
6
with a new stable homological stability theorem, which is the main new result of
1
: this paper. Before explaining the terms in (1.1), let us explain this theorem.
v
i
X 1.1. Statementofthemaintheorem. Inordertostateourhomologicalstability
r results more precisely, we must first introduce a variant of the space BDiff(W)
a which incorporates certain tangent bundle information. As in our earlier paper
[GRW14b] the clearest statements can be made by not fixing a manifold W, but
rather considering all manifolds with a specified boundary at once.
To define this modified classifying space, we fix a map θ :B →BO(2n) classify-
ing a vector bundle θ∗γ over B, and we shall suppose throughout that B is path
2n
connected. We shall call a bundle map TW →θ∗γ a θ-structure on W.
2n
Definition 1.1. Let W be a 2n-dimensional manifold with boundary P, and (cid:96)ˆ :
P
ε1 ⊕TP → θ∗γ be a bundle map. Let Bunθ (TW;(cid:96)ˆ ) denote the space of all
2n n,∂ P
Date:January5,2016.
2010 Mathematics Subject Classification. 57R90,57R15,57R56,55P47.
1
2 SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
bundle maps (cid:96)ˆ: TW → θ∗γ which are equal to (cid:96)ˆ when restricted to P and
2n P
whose underlying map (cid:96):W →B is n-connected. Let
N θ(P,(cid:96)ˆ )=(cid:97)Bunθ (TW;(cid:96)ˆ )//Diff (W),
n P n,∂ P ∂
[W]
where the disjoint union is taken over compact manifolds W, equipped with a
diffeomorphism ∂W ∼= P, one in each diffeomorphism class relative to P. The
topologicalgroupDiff (W)isthegroupofdiffeomorphismsofW restrictingtothe
∂
identity on a neighborhood of ∂W, and “//” denotes the homotopy orbit space.
If M is a cobordism between P and Q such that (M,Q) is (n−1)-connected,
andisequippedwithaθ-structure(cid:96)ˆ restrictingto(cid:96)ˆ overP and(cid:96)ˆ overQ,then
M P Q
there is an induced map
(1.2) −∪ (M,(cid:96)ˆ ):N θ(Q,(cid:96)ˆ )−→N θ(P,(cid:96)ˆ ),
Q M n Q n P
which on the path components corresponding to W is induced by the Diff (W)-
∂
equivariantmapBunθ (TW;(cid:96)ˆ )→Bunθ (T(W∪ M);(cid:96)ˆ )definedbyextending
n,∂ Q n,∂ Q P
bundle maps by (cid:96)ˆ .
M
Ourmainresultwillconcernthehomologicaleffectof(1.2)onceallmanifoldsin
sight have been stabilised by connect-sum with countably many copies of Sn×Sn.
We shall give a precise model for forming this stabilisation in Section 2, but for
now the following explanation of diagram (1.4) below will suffice.
Let p : [0,1]×D2n−1 (cid:44)→ M be an embedding sending {0}×D2n−1 into Q and
{1}×D2n−1 into P. (In particular, P and Q must be non-empty.) We may then
form the connect-sum
H =([0,1]×Q)(cid:93)(Sn×Sn)
Q
insidetheimageoftheembedding[0,1]×D2n−1 (cid:44)→[0,1]×Qinastandardway;we
mayanalogouslyform H. UnlessP andQarepathconnected,thediffeomorphism
P
types of H and H may depend on p and not just P and Q, but we suppress
P Q
this dependence from the notation. Sliding along the thickened path p induces a
diffeomorphism
(1.3) M ∪ ( H)∼=( H)∪ M
P P Q Q
relative to the common boundaries P and Q. The manifolds H and H each
P Q
contain an embedded copy of W =Sn×Sn\int(D2n).
1,1
Definition1.2. Letussaythataθ-structure(cid:96)ˆ:TW →θ∗γ isadmissible ifthere
1,1
is a pair of orientation-preserving embeddings e,f : Sn×Dn (cid:44)→ W ⊂ Sn×Sn
1,1
whose cores e(Sn×{0}) and f(Sn×{0}) intersect transversely in a single point,
such that each of the θ-structures e∗(cid:96)ˆand f∗(cid:96)ˆon Sn×Dn extend to R2n for some
orientation-preserving embeddings Sn×Dn (cid:44)→R2n.
We say that a θ-structure on H is admissible if it restricts to an admissible θ-
P
structureonW andto(cid:96)ˆ on{0}×P. Similarlyfor H. Choosinganadmissible
1,1 P Q
θ-structure (cid:96)ˆ on H, we obtain a θ-structure (cid:96)ˆ ∪(cid:96)ˆ on M ∪ ( H), and
PH P M PH P P
henceviathediffeomorphism(1.3)aθ-structureon( H)∪ M. Restrictedto H
Q Q Q
thisisanadmissibleθ-structure(cid:96)ˆ ,andrestrictedtoM itgivesanewθ-structure
QH
(cid:96)ˆ(cid:48) , which restricts to (cid:96)ˆ(cid:48) over P and (cid:96)ˆ(cid:48) over Q. These manifolds with θ-structure
M P Q
HOMOLOGICAL STABILITY 3
induce maps in the diagram
N θ(Q,(cid:96)ˆ )−∪Q(QH,(cid:96)ˆQH)(cid:47)(cid:47)N θ(Q,(cid:96)ˆ(cid:48) )
n Q n Q
(1.4)
(cid:15)(cid:15)−∪Q(M,(cid:96)ˆM) (cid:15)(cid:15)−∪Q(M,(cid:96)ˆ(cid:48)M)
N θ(P,(cid:96)ˆ ) −∪P(PH,(cid:96)ˆPH)(cid:47)(cid:47)N θ(P,(cid:96)ˆ(cid:48) ),
n P n P
and the diffeomorphism (1.3) induces a homotopy between the two compositions.
Iterating this construction, we obtain an induced map from
hocolim(N θ(Q,(cid:96)ˆ )→N θ(Q,(cid:96)ˆ(cid:48) )→N θ(Q,(cid:96)ˆ(cid:48)(cid:48))→···)
n Q n Q n Q
to the analogous homotopy colimit for (P,(cid:96)ˆ ). This limiting map is denoted −∪
P Q
(M,(cid:96)ˆ(∞)) in the following theorem, which is the main new result of this paper.
M
Theorem 1.3. The map
−∪ (M,(cid:96)ˆ(∞)):hocolimN θ(Q,(cid:96)ˆ(g))−→hocolimN θ(P,(cid:96)ˆ(g))
Q M n Q n P
g→∞ g→∞
induces an isomorphism on homology with all abelian systems of local coefficients.
Remark 1.4. A system of local coefficients L on a space X is called abelian if for
each point x ∈ X the action of π (X,x) on the fibre L(x) factors through an
1
abelian group. In particular, all constant coefficient systems are abelian. If X is
path connected and based then an abelian system of local coefficients is equivalent
to the data of a module over the group ring Z[H (X;Z)].
1
1.2. Stable homology. The homology of the limiting spaces
hocolimN θ(P,(cid:96)ˆ(g))
n P
g→∞
maybededucedfromtheresultsof[GRW14b]. Infact,inSection7weshallrevisit
the results of [GRW14b] and see that the two sections of that paper concerning
“surgery on objects” may be replaced by the homological stability results of the
present paper, by following a line of argument inspired by [Til97]. We shall see
thatinallcasesthehomologyofthelimitingspaceagreeswiththehomologyofthe
infinite loop space of a certain Thom spectrum. This approach using homological
stability does not give simpler or easier proofs of the results of [GRW14b], but it
does work in slightly more generality. For example, we shall be able to remove the
restriction 2n≥6 from [GRW14b].
Recall that we write MTθ for the Thom spectrum of the virtual vector bundle
−θ∗γ over B, and Ω∞MTθ for its infinite loop space.
2n
Theorem 1.5. If N θ(P,(cid:96)ˆ )(cid:54)=∅ then there is a map
n P
hocolimN θ(P,(cid:96)ˆ(g))−→Ω∞MTθ
n P
g→∞
which induces an isomorphism on homology, and is in fact acyclic.
In fact, we will construct a specific acyclic map, using a cobordism category C .
θ
The space N θ(P,(cid:96)ˆ ) will be modelled as a subspace of the space of morphisms
n P
C (∅,(P,(cid:96)ˆ )) in C , and hence comes with a canonical map
θ P θ
N θ(P,(cid:96)ˆ )⊂C (∅,(P,(cid:96)ˆ ))−→Ω BC ,
n P θ P [∅,(P,(cid:96)ˆP)] θ
where BC denotes the classifying space (geometric realisation of the nerve) and
θ
Ω denotes the space of paths between the points corresponding to the ob-
[∅,(P,(cid:96)ˆP)]
jects ∅ and (P,(cid:96)ˆ ). The theorem of Galatius–Madsen–Tillmann–Weiss [GMTW09]
P
4 SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
providesaweakequivalencefromthisspaceofpaths,whennon-empty,toΩ∞MTθ.
This is the map which the theorem is about.
Remark 1.6. A map f : X → Y is acyclic if for any system of local coefficients L
on Y the induced map f : H (X;f∗L) → H (Y;L) is an isomorphism. This is
∗ ∗ ∗
equivalent to the homotopy fibre of f over any point of Y being acyclic, i.e. having
the integral homology of a point.
In the statement of Theorem 1.5 the target has abelian fundamental group.
Thus every system of local coefficients is abelian, and so the map being acyclic is
equivalent to inducing an isomorphism on homology with all abelian systems of
local coefficients.
Theorem1.5,andtheslightlymoregeneralTheorem7.3,improveson[GRW14b,
Theorem 1.8] in a few ways. Firstly, it applies to manifolds of all even dimensions,
and not just dimensions 2n≥6; secondly, it does not require that θ be “spherical”
(a technical condition introduced in [GRW14b]); thirdly, it requires a simpler form
of stabilisation (only copies of W have to be glued on, rather than a “universal
1,1
θ-end”); fourthly, the conclusion is that the map is acyclic rather than merely a
homology equivalence.
1.3. Finite genus and closed manifolds. We can combine Theorem 1.3 with
our earlier work [GRW14a] to obtain results about the homology of the spaces
N θ(P;(cid:96)ˆ ) before stabilising by W . If we write Mθ(W,(cid:96)ˆ ) ⊂ N θ(Q,(cid:96)ˆ ) for
n P 1,1 n W n Q
the path component containing (W,(cid:96)ˆ ), then the map (1.2) restricts to a map
W
(1.5) −∪ (M,(cid:96)ˆ ):Mθ(W,(cid:96)ˆ )−→Mθ(W(cid:48),(cid:96)ˆ ),
Q M n W n W(cid:48)
where W(cid:48) = W ∪ M and (cid:96)ˆ is the bundle map obtained by gluing (cid:96)ˆ and (cid:96)ˆ .
Q W(cid:48) W M
Recall from [GRW14a] that the genus of a θ-manifold is defined as
(cid:26) (cid:12) (cid:27)
gθ(W,(cid:96)ˆW)=max g ∈N (cid:12)(cid:12) there are g disjoint copies of W1,1 in W,
(cid:12) each with admissible θ-structure
and the stable genus is defined as
g¯θ(W,(cid:96)ˆ )=max{gθ(W(cid:93)W ,(cid:96)ˆ(k))−k | k ∈N},
W k,1 W
where W(cid:93)W is the manifold obtained from W by cutting out a disc and forming
k,1
theboundaryconnectedsumwithkcopiesofW . Thisgluedmanifoldisequipped
1,1
withaθ-structure(cid:96)ˆ(k)obtainedbyextendingtherestrictionof(cid:96)ˆ byanyadmissible
W W
structures on the W . In Section 8 we shall explain how to deduce the following
1,1
two finite-genus results from the results in this paper and [GRW14a]. (In fact we
shall deduce slightly more general results which impliy the corollaries but which
also imply some homological stability for H (N θ(−);Z).)
0 n
Corollary 1.7. Assume 2n ≥ 6, B is simply-connected, and (M,(cid:96)ˆ ) is a θ-
M
cobordism between P and Q such that (M,Q) is (n−1)-connected, and denote by
Mθ(W,(cid:96)ˆ )⊂N θ(Q,(cid:96)ˆ )thecomponentcontaining(W,(cid:96)ˆ ). Writeg =g¯θ(W,(cid:96)ˆ ),
n W n Q W W
and let L be an abelian coefficient system on the target of the map (1.5). Then the
induced map
(−∪ (M,(cid:96)ˆ )) :H (Mθ(W,(cid:96)ˆ );(−∪ M)∗L)−→H (Mθ(W(cid:48),(cid:96)ˆ );L)
Q M ∗ i n W Q i n W(cid:48)
is an isomorphism, provided one of the following three assumptions hold:
(i) 2i≤g−3, L is constant, and θ is spherical;
(ii) 3i≤g−4 and L is constant;
(iii) 3i≤g−4, g ≥5, and either g ≥7 or θ is spherical.
HOMOLOGICAL STABILITY 5
Corollary 1.8. There is a map
Mθ(W,(cid:96)ˆ )−→Ω∞MTθ
n W 0
which, under the same conditions on n and B as in Corollary 1.7, is acyclic in
degrees satisfying 3∗ ≤ g¯θ(W,(cid:96)ˆ )−4, and induces an isomorphism on integral
W
homology in degrees satisfying 2∗≤g¯θ(W,(cid:96)ˆ )−3 if in addition θ is spherical.
W
As in Theorem 1.5 and the discussion following it, Corollary 1.8 concerns a
specificmapintoacomponentofapathspaceofBC ,whichby[GMTW09]maybe
θ
identifiedwiththebasepointcomponentΩ∞MTθoftheinfiniteloopspaceΩ∞MTθ.
0
1.4. Other tangential structures. In the definition of N θ(P,(cid:96)ˆ ), if the space
n P
Bunθ (TW;(cid:96)ˆ ), which consists of n-connected bundle maps with fixed boundary
n,∂ P
condition,isreplacedbythespaceofall(notnecessarilyn-connected)bundlemaps
with fixed boundary condition then the results of this paper may still be used to
deduce the homology of the resulting moduli space in a stable range. Again we
state the result one path component at a time, and for simplicity we shall restrict
attention to closed manifolds in this introduction.
Letθ(cid:48) :B(cid:48) →BO(2n)beamap, andletW beaclosed2n-dimensionalmanifold
and (cid:96)ˆ(cid:48) :TW →(θ(cid:48))∗γ a θ(cid:48)-structure. We shall write Bunθ(cid:48)(TW) for the space of
W
all (not necessarily n-connected) θ(cid:48)-structures and
Mθ(cid:48)(W,(cid:96)ˆ )⊂Bunθ(cid:48)(TW)//Diff(W)
W
for the path component containing the image of (cid:96)ˆ(cid:48) . Let (cid:96)(cid:48) :W →B(cid:48) be the map
W W
underlying (cid:96)ˆ(cid:48) and let W →B →B(cid:48) be a factorisation of (cid:96)(cid:48) into an n-connected
W W
map (cid:96) : W → B and an n-co-connected fibration u : B → B(cid:48). Let us write
W
θ = θ(cid:48) ◦ u : B → BO(2n). In Section 9 below, we explain how to deduce the
following result from Corollary 1.8.
Corollary 1.9. In this situation the topological monoid hAut(u), consisting of
weak equivalences B → B commuting with the map u : B → B(cid:48), acts on the space
Ω∞MTθ and there is a map
Mθ(cid:48)(W,(cid:96)ˆ(cid:48) )−→(Ω∞MTθ)//hAut(u).
W
Consideredasamaptothepathcomponentwhichithits,if2n≥6andW issimply-
connectedthenthismapisacyclicindegreessatisfying3∗≤g¯θ(cid:48)(W,(cid:96)ˆ(cid:48) )−4,andin-
W
ducesanisomorphisminintegralhomologyindegreessatisfying2∗≤g¯θ(cid:48)(W,(cid:96)ˆ(cid:48) )−3
W
if in addition θ(cid:48) is spherical.
TheproofgiveninSection9againconcernsaspecificmap,andwewillalsotreat
the case ∂W (cid:54)=∅.
Taking θ(cid:48) = Id : BO(2n) → BO(2n) and letting W (cid:96)→W B →u BO(2n) be a
Moore–Postnikov n-stage factorisation of a Gauss map (cid:96)(cid:48) : W → BO(2n), the
W
space Mθ(cid:48)(W,(cid:96)ˆ(cid:48) ) is a model for BDiff(W) and we obtain a map
W
(1.6) BDiff(W)−→(Ω∞MTθ)//hAut(u)
which, as long as 2n ≥ 6 and W is simply-connected, is a homology isomorphism
(respectively acyclic) in a range of degrees depending on g(W), onto the path
component which it hits.
Similarly we can take θ(cid:48) : BSO(2n) → BO(2n) to be the orientation cover and
letW (cid:96)→W B u→+ BSO(2n)beaMoore–Postnikovn-stagefactorisationofanoriented
Gauss map (cid:96)(cid:48) : W → BSO(2n). Then the space Mθ(cid:48)(W,(cid:96)ˆ(cid:48) ) is a model for
W W
BDiff+(W) and we obtain a map
BDiff+(W)−→(Ω∞MTθ)//hAut(u+)
6 SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
which, as long as 2n ≥ 6 and W is simply-connected, is a homology isomorphism
(respectively acyclic) in a range of degrees depending on g(W), onto the path
component which it hits.
1.5. Comparingclosedmanifolds. IfW isaclosed2n-manifoldthentheMoore–
Postnikov n-stage of the Gauss maps of W and W(cid:93)g(Sn ×Sn) define the same
tangential structure θ :B →BO(2n) up to homotopy, as these manifolds have the
same (n−1)-skeleta, and their n-skeleta differ by the addition of n-cells on which
the tangent bundle is trivial. Thus (1.6) gives maps
(cid:16) (cid:17)
BDiff(W)−→ Ω∞ MTθ //hAut(θ)
(W)
(cid:16) (cid:17)
BDiff(W(cid:93)g(Sn×Sn))−→ Ω∞ MTθ //hAut(θ)
(W(cid:93)g(Sn×Sn))
which, if 2n ≥ 6 and W is simply-connected, are both isomorphisms on integral
homology in degrees satisfying 2∗ ≤ g(W)−3. The targets of these two maps
are defined as the two (distinct, if g >0) path components of (Ω∞MTθ)//hAut(θ)
hit by the map. If the homotopy orbit space (Ω∞MTθ)//hAut(θ) were an infinite
loopspacewecouldarguethatallpathcomponentswouldbeweaklyequivalentand
hencededucethatthespacesBDiff(W)andBDiff(W(cid:93)g(Sn×Sn))havecanonically
isomorphic homology in degrees satisfying 2∗ ≤ g(W)−3, even though there are
nodirectmapsbetweenthesetwospaces. FormostW thehomotopyorbitspaceis
notaninfiniteloopspace,butneverthelesswehavethefollowingcomparisonresult
after inverting finitely many prime numbers.
Theorem 1.10. LetW beaclosedsimply-connectedmanifoldofdimension2n≥6
and let W(cid:48) = W(cid:93)g(Sn×Sn) for some g ∈ N. Assume χ(W) and χ(W(cid:48)) are both
non-zero, and let R⊂Q be the smallest subring containing χ(W) and χ(W(cid:48)). Then
χ(W(cid:48)) χ(W)
the spaces BDiff(W) and BDiff(W(cid:48)) have isomorphic homologies with coefficients
inanyR-module,indegreessatisfying2∗≤g(W)−3. Thisisomorphismisinduced
by a zig-zag of maps of spaces.
Similarly for BDiff+(W) and BDiff+(W(cid:48)).
In particular, if both Euler characteristics are non-zero then these spaces have
thesamerationalhomologyinthisrangeofdegrees,andiftheEulercharacteristics
have the same p-adic valuation then these spaces have the same p-local homology
in this range of degrees.
In a companion paper [GRW] we shall show that the condition on the Euler
characteristicscannotberemoved: ingeneralBDiff(W)andBDiff(W(cid:93)g(Sn×Sn))
do not have the same integral homology in low degrees.
1.6. Acknowledgements. S. Galatius was partially supported by NSF grants
DMS-1105058 and DMS-1405001, as well as the Danish National Research Foun-
dation through the Centre for Symmetry and Deformation (DNRF92). O. Randal-
Williams was partially supported by the Herchel Smith Fund and EPSRC grant
EP/M027783/1.
HOMOLOGICAL STABILITY 7
Contents
1. Introduction and statement of results 1
1.1. Statement of the main theorem 1
1.2. Stable homology 3
1.3. Finite genus and closed manifolds 4
1.4. Other tangential structures 5
1.5. Comparing closed manifolds 6
1.6. Acknowledgements 6
2. Definitions, and statement of the main theorem 8
2.1. Cobordism categories 8
2.2. Standard θ-structures 9
2.3. Stable homological stability 10
2.4. Models for moduli spaces of manifolds 11
2.5. Elementary simplifications of Theorem 2.14 13
3. Proof of Theorem 2.14: stability for W 15
1,1
4. Proof of Theorem 2.14: stability for handles of index n 16
4.1. Support and interchange of support 17
4.2. Constructing auxiliary cobordisms 17
4.3. A semi-simplicial resolution 20
4.4. Resolving composition with M ◦r(M) 21
5. Contractibility of the higher-dimensional arc complex 26
6. Proof of Theorem 2.14: stability for handles of index n<k <2n 34
6.1. Constructing auxiliary cobordisms 34
6.2. A semi-simplicial resolution 35
6.3. Resolving composition with M 37
7. Stable homology and group-completion 38
7.1. The group-completion argument 39
7.2. Proof of Theorem 7.3 43
8. Finite genus and stability for closed manifolds 45
8.1. Finite genus and homological stability 45
8.2. Proof of Corollary 1.7 for P (cid:54)=∅ 46
8.3. Proof of Corollary 1.7 in general 47
8.4. Stable homology and proof of Corollary 1.8 49
9. General tangential structures 49
10. Comparing closed manifolds 52
Appendix A. Homology equivalences and local coefficients 57
A.1. Local coefficient systems 57
A.2. Invariance properties for abelian coefficients 59
A.3. Ladder diagrams and abelian equivalences 61
A.4. The generalised group completion theorem with local coefficients 62
References 64
8 SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
2. Definitions, and statement of the main theorem
Inthefollowingweshallsupposethat n>0. Weemphasisethatweallow n=1
and n=2, and we will point out when these cases need special treatment.
2.1. Cobordism categories. It is convenient to state the precise version of The-
orem 1.3 using the language of cobordism categories; these are also useful for
the proof of Theorem 1.3, and will be essential for the proof of Theorem 1.5.
See [GMTW09, GRW10, GRW14b] for further background on these categories.
ThroughoutthepaperweshallworkwithmanifoldsinsideR×R∞,andwillwritee
0
forthebasisvectorofthefirstcopyofR. Apartfromthis,wewillwritee ,e ,...,e
1 2 k
for the basis vectors of Rk.
Definition 2.1. Let C be the (non-unital) category with objects given by
θ
(i) a (2n−1)-dimensional closed submanifold P ⊂R∞, and
(ii) a bundle map (cid:96)ˆ :ε1⊕TP →θ∗γ.
P
A morphism from (P,(cid:96)ˆ ) to (Q,(cid:96)ˆ ) is given by
P Q
(i) an s∈(0,∞),
(ii) a2n-dimensionalsubmanifoldW ⊂[0,s]×R∞ whichforsomeε>0intersects
([0,ε)×R∞)∪((s−ε,s]×R∞)
in ([0,ε)×P)∪((s−ε,s]×Q),
(iii) a bundle map (cid:96)ˆ :TW →θ∗γ which restricts to (cid:96)ˆ on {0}×P and to (cid:96)ˆ on
W P Q
{s}×Q.
The composition of (s,W,(cid:96)ˆ ) : (P,(cid:96)ˆ ) (cid:32) (Q,(cid:96)ˆ ) and (s(cid:48),W(cid:48),(cid:96)ˆ ) : (Q,(cid:96)ˆ ) (cid:32)
W P Q W(cid:48) Q
(R,(cid:96)ˆ ) is given by
R
(s+s(cid:48),W ∪(W(cid:48)+s·e ),(cid:96)ˆ ∪(cid:96)ˆ ):(P,(cid:96)ˆ )(cid:32)(R,(cid:96)ˆ ).
0 W W(cid:48) P R
This does not quite describe a category, as there are no identity morphisms, but it
willbeenoughtoworkinsuchanon-unitalcategory. WemakeC intoatopological
θ
(non-unital) category using the evident isomorphism with the topological (non-
unital)categorydescribedin[GRW14b,Definition2.6]. Inparticular,thetopology
on the morphism space C ((P,(cid:96)ˆ ),(Q,(cid:96)ˆ )) is that of a subquotient of
θ P Q
 
(cid:97)
(0,∞)× Emb(W,[0,1]×R∞)×Bunθ(TW),
[W]
where W ranges over the set of all cobordisms from P to Q, one in each diffeomor-
phism class.
Definition 2.2. Let L ⊂ (−∞,0] × R∞−1 be a compact (2n − 1)-dimensional
submanifold which near {0} × R∞−1 coincides with (−∞,0] × ∂L, and let (cid:96)ˆ :
L
ε1⊕TL→θ∗γ be a θ-structure which is a product near the boundary.
2n
Let C ⊂C be the subcategory with
θ,L θ
(i) objects those (P,(cid:96)ˆ ) such that P ∩((−∞,0]×R∞−1)=L and (cid:96)ˆ | =(cid:96)ˆ ,
P P L L
(ii) morphisms those (s,W,(cid:96)ˆ ) such that
W
W ∩([0,s]×((−∞,0]×R∞−1))=[0,s]×L
and (cid:96)ˆ restricts to the bundle map T([0,s]×L)→ε1⊕TL→(cid:96)ˆL θ∗γ .
W 2n
Weshallgenerallyabbreviateθ-manifoldsbythenameoftheunderlyingsmooth
manifold, say X, and write (cid:96)ˆ for its θ-structure. Using this convention, for any
X
HOMOLOGICAL STABILITY 9
object P ∈ C and any s ∈ (0,∞) there is an morphism (s,[0,s] × P), having
θ
underlying manifold [0,s]×P ⊂[0,s]×R∞ and θ-structure given by
(cid:96)ˆ :T([0,s]×P)−→ε1⊕TP −(cid:96)ˆ→P θ∗γ .
[0,s]×P 2n
We shall call such morphisms “cylindrical”. We can make the analogous construc-
tion for objects in C .
θ,L
Remark 2.3. Although the categories C and C do not have any identity mor-
θ θ,L
phisms, foreachobjectP ineithercategorythecylindricalmorphism(s,[0,s]×P)
has the property that the self-maps of C (P,Q) and C (Q,P) given by pre- and
θ θ
post-composingwithitishomotopictotheidentity. Thuswewillsaythatacobor-
dism M : P (cid:32) Q is an isomorphism in one of these categories if there is an
N : Q (cid:32) P such that M ◦N and N ◦M are in the same path component as
cylindrical morphisms in the space of morphisms of the category.
Themostimportantexamplesofisomorphismswillbecobordismswhoseunder-
lying manifold is diffeomorphic to a cylinder, but with non-cylindrical θ-structure
or embedding.
Definition 2.4 ([GRW14b]). LetCn−1 ⊂C bethesubcategoryhavingthesame
θ,L θ,L
objects, and having those morphisms (s,M):P (cid:32)Q for which the pair (M,Q) is
(n−1)-connected.
A crucial feature of the categories C is that they do not depend on L, but
θ,L
rather only on ∂L: if L(cid:48) is another collared θ-manifold which is equal to L near its
∼
boundary, thenweobtainanisomorphismoftopologicalcategoriesC →C by
θ,L θ,L(cid:48)
cutting out L and gluing in L(cid:48) (and similarly on morphisms). In order to make use
of this, we make the following definition.
Definition 2.5. If P is an object of C , let P◦ ⊂ [0,∞) × R∞−1 be the θ-
θ,L
manifold obtained by cutting out int(L). Similarly, if (s,W) is a morphism in
C , let W◦ ⊂ [0,s]×[0,∞)×R∞−1 be the θ-cobordism obtained by cutting out
θ,L
[0,s]×int(L).
Let C be the (non-unital) category defined by
θ,∂L
Ob(C )={(P◦,(cid:96)ˆ )|(P,(cid:96)ˆ )∈Ob(C )}
θ,∂L P◦ P θ,L
Mor(C )={(s,W◦,(cid:96)ˆ )|(s,W,(cid:96)ˆ )∈Mor(C )}
θ,∂L W◦ W θ,L
and made into a topological category by insisting that the functor C → C
θ,L θ,∂L
given by X (cid:55)→X◦ is an isomorphism of topological categories.
WeshallgenerallywriteP,Q,...forobjectsandW,M,...formorphismsofC
θ,∂L
ratherthanC ,andreservethe(−)◦ notationforwhenwearedirectlycomparing
θ,L
elements in C with their corresponding elements in C .
θ,L θ,∂L
Definition 2.6. Let Cn−1 ⊂ C be the subcategory having the same objects,
θ,∂L θ,∂L
and having those morphisms (s,M◦) : P◦ (cid:32) Q◦ for which the pair (M◦,Q◦) is
(n−1)-connected.
Remark 2.7. If(s,M◦):P◦ (cid:32)Q◦ ∈Cn−1 thenthecobordismM =([0,s]×L)∪M◦
θ,∂L
is (n−1)-connected relative to Q=L∪Q◦, which defines a functor Cn−1 →Cn−1.
θ,∂L θ,L
This is not in general an isomorphism of categories, but if (L,∂L) is (n − 1)-
connected then it is. (See [GRW14b, Lemma 7.2] for a related discussion.)
2.2. Standard θ-structures. Let θ :B →BO(2n) be a tangential structure, and
recallthatweinsistthatBispathconnected. Chooseonceandforalla“basepoint”
θ-structure onthe vector space R2n, i.e.a bundle map τ :R2n →θ∗γ (depending
2n
on the orientability of θ∗γ , there are one or two possible such choices, up to
2n
10 SØRENGALATIUSANDOSCARRANDAL-WILLIAMS
homotopy). Thisinducesacanonicalθ-structure(cid:96)ˆτ onanyframed2n-or(2n−1)-
X
manifold X.
Definition 2.8. Thestandard framing ξ onSn×Dn istheframinginduced
Sn×Dn
by the embedding
Sn×Dn −→Rn+1×Rn−1 =R2n
(x;y1,y2,...,yn)(cid:55)−→(2e−y21x;y2,...,yn).
This framing induces a θ-structure (cid:96)ˆτ , and we say that a θ-structure on Sn×
Sn×Dn
Dn is standard if it is homotopic to (cid:96)ˆτ .
Sn×Dn
Definition 2.9. Let W =Sn×Sn\int(D2n) be obtained by removing an open
1,1
discinsideDn×Dn,theproductofthetwoupperhemispheres. Aθ-structure(cid:96)ˆon
+ +
W is standard if both θ-structures e¯∗(cid:96)ˆand f¯∗(cid:96)ˆon Sn×Dn are standard, where
1,1
e¯and f¯are the embeddings defined by
e¯:Sn×Dn −→W ⊂Sn×Sn ⊂Rn+1×Rn+1
1,1
(cid:16) (cid:113) (cid:17)
(x,y)(cid:55)−→ x;y,− 1−|y|2
2 2
and
f¯:Sn×Dn −→W ⊂Sn×Sn ⊂Rn+1×Rn+1
1,1
(cid:16) (cid:113) (cid:17)
(x,y)(cid:55)−→ −y,− 1−|y|2;x ,
2 2
Remark 2.10. In Definition1.2 we said that a θ-structure (cid:96)ˆon W was admissible
1,1
if there are orientation-preserving embeddings e,f : Sn ×Dn → W with cores
1,1
intersectingtransverselyinonepoint,suchthateachoftheθ-structurese∗(cid:96)ˆandf∗(cid:96)ˆ
onSn×Dn extendtoR2n forsomeorientation-preservingembeddingsSn×Dn (cid:44)→
R2n. In particular, as e¯(Sn×Dn)∪f¯(Sn×Dn) ⊂ W is an isotopy equivalence
1,1
it follows that if (cid:96)ˆis admissible then there is an embedding φ : W (cid:44)→ W such
1,1 1,1
that φ∗(cid:96)ˆis standard.
The embeddings e¯ and f¯are orientation-preserving and the union of their im-
ages is isotopy equivalent to W . Therefore there exists a framing ξ of W
1,1 W1,1 1,1
such that e¯∗ξ and f¯∗ξ are each homotopic to ξ , so the associated
W1,1 W1,1 Sn×Dn
θ-structure (cid:96)ˆτ is standard.
W1,1
Lemma 2.11. The space of standard θ-structures on W fixed on a disc in the
1,1
boundary is path-connected.
Proof. This is [GRW14a, Lemma 7.7], under a choice of identification of W with
1,1
the manifold H of that paper. (cid:3)
2.3. Stable homological stability. In this section we shall formulate our stable
homological stability theorem, which is a strengthening of Theorem 1.3.
Definition 2.12. We say that a composable sequence of cobordisms
K| K(cid:32)|[0,1] K| K(cid:32)|[1,2] K| K(cid:32)|[2,3] K| (cid:32)···
0 1 2 3
in C is a θ-end in C if each cobordism K| satisfies
θ,∂L θ,∂L [i,i+1]
(i) it is (n−1)-connected relative to both K| and K| , and
i i+1
(ii) it contains an embedded copy of W with standard θ-structure, in a path
1,1
component intersecting ∂L.