Deformation of Singularities and the Homology of Intersection Spaces

0.34 MB·English

by  Markus Banagl

DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF
INTERSECTION SPACES
MARKUS BANAGL AND LAURENTIU MAXIM
1
1 Abstract. While intersection cohomology is stable under small resolutions, both ordi-
0
naryandintersectioncohomologyareunstableundersmoothdeformationofsingularities.
2
Forcomplexprojectivealgebraichypersurfaceswithanisolatedsingularity,weshowthat
n
the first author’s cohomology of intersection spaces is stable under smooth deformations
a
J in all degrees except possibly the middle, and in the middle degree precisely when the
monodromy action on the cohomology of the Milnor fiber is trivial. In many situations,
5
2 the isomorphism is shown to be a ring homomorphism induced by a continuous map.
This is used to show that the rationalcohomologyof intersectionspaces can be endowed

] with a mixed Hodge structure compatible with Deligne’s mixed Hodge structure on the
T
ordinary cohomology of the singular hypersurface.
A
.
h
t
a
m
Contents
[

1. Introduction 1
1
v 2. Background on Intersection Spaces 5
3 3. Background on Hypersurface Singularities 9
8
4. Deformation Invariance of the Homology of Intersection Spaces 13
8
4 5. Maps from Intersection Spaces to Smooth Deformations 19
.
1 6. Deformation of Singularities and Intersection Homology 27
0 7. Higher-Dimensional Examples: Conifold Transitions 29
1
References 31
1
:
v
i
X
r 1. Introduction
a
Given a singular complex algebraic variety V, there are essentially two systematic geo-
metric processes for removing the singularities: one may resolve them, or one may pass to
a smooth deformation of V. Ordinary homology is highly unstable under both processes.
This is evident from duality considerations: the homology of a smooth variety satisfies
Poincar´e duality, whereas the presence of singularities generally prevents Poincar´e duality.
Date: January 26, 2011.
Keywords andphrases. Singularities,projectivehypersurfaces,smoothdeformations,Poincar´eduality,
intersection homology, Milnor fibration, mixed Hodge structures, mirror symmetry.
The first author was in part supported by a research grant of the Deutsche Forschungsgemeinschaft.
The second author was partially supported by NSF-1005338.
1
2 MARKUSBANAGL ANDLAURENTIUMAXIM
Goresky and MacPherson’s middle-perversity intersection cohomology IH∗(V;Q), as well
as Cheeger’s L2-cohomology H∗ (V) do satisfy Poincar´e duality for singular V; thus it
(2)
makes sense to ask whether these theories are stable under the above two processes. The
answer is that both are preserved under so-called small resolutions. Not every variety
possesses a small resolution, though it does possess some resolution. Both IH∗ and H∗
(2)
are unstable under smooth deformations. For projective hypersurfaces with isolated sin-
gularities, the present paper answers positively the question: Is there a cohomology theory
for singular varieties, which is stable under smooth deformations? Note that the smallness
condition on resolutions needed for the stability of intersection cohomology suggests that
the class of singularities for which such a deformation stable cohomology theory exists
must also be restricted by some condition.
Let p¯ be a perversity in the sense of intersection homology theory. In [Ba10], the first
author introduced a homotopy-theoretic method that assigns to certain types of real n-
dimensional stratified topological pseudomanifolds X CW-complexes
Ip¯X,
the perversity-p¯ intersection spaces of X, such that for complementary perversities p¯and
q¯, there is a Poincar´e duality isomorphism
Hi(Ip¯X;Q) = H (Iq¯X;Q)
∼ n−i
when X is compact and oriented. This method is in particular applicable to complex
e e
algebraic varieties V with isolated singularities, whose links are simply connected. The
latter is a sufficient, but not a necessary condition. If V is an algebraic variety, then Ip¯V
will in general not be algebraic anymore. If p¯= m¯ is the lower middle perversity, we will
briefly write IX for Im¯X. The groups
HI∗(X;Q) = H∗(Ip¯X;Q)
p¯
define a new cohomology theory for stratified spaces, usually not isomorphic to inter-
section cohomology IH∗(X;Q). This is already apparent from the observation that
p¯
HI∗(X;Q) is an algebra under cup product, whereas it is well-known that IH∗(X;Q)
p¯ p¯
cannot generally, for every p¯, be endowed with a p¯-internal algebra structure. Let us put
HI∗(X;Q) = H∗(IX;Q).
It was pointed out in [Ba10] that in the context of conifold transitions, the ranks of
HI∗(V;Q) for a singular conifold V agree with the ranks of H∗(V ;Q), where V is a
s s
nearby smooth deformation of V; see the table on page 199 and Proposition 3.6 in loc.
cit. The main result, Theorem 4.1, of the present paper is the following Stability Theorem.
Theorem. Let V be a complex projective hypersurface of complex dimension n = 2 with
6
one isolated singularity and let V be a nearby smooth deformation of V. Then, for all
s
i < 2n and i = n, we have
6
Hi(V ;Q) = HIi(V;Q).
s ∼
e e
SINGULARITIES AND INTERSECTION SPACE HOMOLOGY 3
Moreover,
Hn(V ;Q) = HIn(V;Q)
s ∼
if, and only if, the monodromy operator acting on the cohomology of the Milnor fiber of
the singularity is trivial.
The case of a surface n = 2 is excluded because a general construction of the intersection
spaceinthiscaseispresently notavailable. However, thetheoryHI∗(V;R)hasadeRham
description [Ba11] by a certain complex of global differential forms on the top stratum of
V, which does not require that links be simply connected. Using this description of HI∗,
the theorem can be extended to the surface case. The description by differential forms is
beyond the scope of this paper and will not be further discussed here.
Let us illustrate the Stability Theorem with a simple example. Consider the equation
y2 = x(x 1)(x s)
− −
(or its homogeneous version v2w = u(u w)(u sw), defining a curve in CP2), where
− −
the complex parameter s is constrained to lie inside the unit disc, s < 1. For s = 0, the
| | 6
equation defines an elliptic curve V , homeomorphic to a 2-torus T2. For s = 0, a local
s
isomorphism
V = y2 = x2(x 1) η2 = ξ2
{ − } −→ { }
near the origin is given by ξ = xg(x), η = y, with g(x) = √x 1 analytic and nonzero
−
near 0. The equation η2 = ξ2 describes a nodal singularity at the origin in C2, whose
link is ∂I S1, two circles. Thus V is homeomorphic to a pinched T2 with a meridian
×
collapsed to a point, or, equivalently, a cylinder I S1 with coned-off boundary. The
×
ordinary homology group H (V;Z) has rank one, generated by the longitudinal circle.
1
The intersection homology group IH (V;Z) agrees with the intersection homology of the
1
normalization S2 of V:
IH (V;Z) = IH (S2;Z) = H (S2;Z) = 0.
1 1 1
Thus, as H (V ;Z) = H (T2;Z) = Z Z, neither ordinary homology nor intersection
1 s 1
homology remains invariant under the s⊕moothing deformation V ; V . The middle per-
s
versity intersection space IV of V is a cylinder I S1 together with an interval, whose
×
one endpoint is attached to a point in 0 S1 and whose other endpoint is attached to
{ }×
a point in 1 S1. Thus IV is homotopy equivalent to the figure eight and
{ }×
H (IV;Z) = Z Z,
1
⊕
which does agree with H (V ;Z). Several other examples are worked out throughout the
1 s
paper, including a reducible curve, a Kummer surface and quintic threefolds with nodal
singularities.
We can be more precise about the isomorphisms of the Stability Theorem. Given V,
there is a canonical map IV V, and given a nearby smooth deformation V of V, one
s
→
has the specialization map V V. In Proposition 5.1, we construct a map IV V
s s
→ →
4 MARKUSBANAGL ANDLAURENTIUMAXIM
such that IV V V is a factorization of IV V. The map IV V induces the
s s
→ → → →
isomorphisms of the Stability Theorem. It follows in particular that one has an algebra
isomorphism HI∗(V;Q) = H∗(V ;Q) (indegrees less than2n, andfortrivial monodromy).
∼ s
We use this geometrically induced isomorphism to show that under the hypotheses of the
proposition, HeI∗(V;Q) canebe equipped with a mixed Hodge structure, so that IV V
→
induces a homomorphism of mixed Hodge structures on cohomology (Corollary 5.3).
The relationship between IH∗ and HI∗ is very well illuminated by mirror symmetry,
which tends to exchange resolutions and deformations. It is for instance conjectured in
[Mor99] that the mirror of a conifold transition, which consists of a deformation s
→
0 (degeneration smooth to singular) followed by a small resolution, is again a conifold
transition, but performed in the reverse direction. This observation strongly suggests that
since there is a theory IH∗ stable under small resolutions, there ought to be a mirror
theory HI∗ stable under certain “small” deformations. This is confirmed by the present
paper and by the results of Section 3.8 in [Ba10], where it is shown that if V◦ is the mirror
of a conifold V, both sitting in mirror symmetric conifold transitions, then
rkIH (V) = rkHI (V◦)+rkHI (V◦)+2,
3 2 4
rkIH (V◦) = rkHI (V)+rkHI (V)+2,
3 2 4
rkHI (V) = rkIH (V◦)+rkIH (V◦)+2, and
3 2 4
rkHI (V◦) = rkIH (V)+rkIH (V)+2.
3 2 4
In the same spirit, the well-known fact that the intersection homology of a complex variety
V is a vector subspace of the ordinary homology of any resolution of V is “mirrored” by
our result proved in Theorem 5.2 below, stating that the intersection space homology
HI (V) is a subspace of the homology H (V ) of any smoothing V of V.
∗ ∗ s s
Since mirror symmetry is a phenomenon that arose originally in string theory, it is not
surprising that the theories IH∗, HI∗ have a specific relevance for type IIA, IIB string
theories, respectively. While IH∗ yields the correct count of massless 2-branes on a coni-
fold in type IIA theory, the theory HI∗ yields the correct count of massless 3-branes on a
conifold in type IIB theory. These are Propositions 3.6, 3.8 and Theorem 3.9 in [Ba10].
The Euler characteristics χ of IH∗ and HI∗ are compared in Corollary 4.6; the result
is seen to be consistent with the formula
◦
χ(H (V)) χ(IH (V)) = 1 χ(IH (coneL )) ,
∗ ∗ ∗ x
− −
x∈SXing(V)(cid:16) (cid:17)
◦
where coneL is the open cone on the link L of the singularity x, obtained in [CMS08].
x x
The behavior of classical intersection homology under deformation of singularities is dis-
cussed from a sheaf-theoretic viewpoint in Section 6. Proposition 6.1 observes that the
perverse self-dual sheaf ψ (Q )[n], where ψ is the nearby cycle functor of a smooth de-
π X π
forming family π : X S with singular fiber V = π−1(0), is isomorphic in the derived
→
category of V to the intersection chain sheaf IC if, and only if, V is nonsingular. The
V
hypercohomology of ψ (Q )[n] computes the cohomology of the general fiber V and the
π X s
SINGULARITIES AND INTERSECTION SPACE HOMOLOGY 5
hypercohomology of IC computes IH∗(V).
V
Finally, the phenomena described in this paper seem to have a wider scope than hyper-
surfaces. The conifolds and Calabi-Yau threefolds investigated in [Ba10] were not assumed
to be hypersurfaces, nevertheless HI∗ was seen to be stable under the deformations arising
in conifold transitions.
Notation. Rational homology will be denoted by H (X),IH (X),HI (X), whereas
∗ ∗ ∗
integral homology will be written as H (X;Z),IH (X;Z),HI (X;Z). The linear dual of
∗ ∗ ∗
a rational vector space W will be written as W∗ = Hom(W,Q). For a topological space
X, H (X) and H∗(X) denote reduced (rational) homology and cohomology, respectively.
∗
2. Background on Intersection Spaces
e e
In [Ba10], the first author introduced a method that associates to certain classes of
stratified pseudomanifolds X CW-complexes
Ip¯X,
the intersection spaces of X, where p¯is a perversity in the sense of Goresky and MacPher-
son’s intersection homology, such that the ordinary (reduced, rational) homology H (Ip¯X)
∗
satisfies generalized Poincar´e duality when X is closed and oriented. The resulting ho-
mology theory X ; HIp¯(X) = H (Ip¯X) is neither isomorphic to intersection hoemology,
∗ ∗
whichwewillwriteasIHp¯(X),nor(forrealcoefficients) linearlydualtoL2-cohomologyfor
∗
Cheeger’s conical metrics. The Goresky-MacPherson intersection chaincomplexes ICp¯(X)
∗
are generally not algebras, unless p¯ is the zero-perversity, in which case ICp¯(X) is essen-
∗
tially the ordinary cochain complex of X. (The Goresky-MacPherson intersection product
raises perversities in general.) Similarly, the differential complex Ω∗ (X) of L2-forms on
(2)
the top stratum is not an algebra under wedge product of forms. Using the intersection
space framework, the ordinary cochain complex C∗(Ip¯X) of Ip¯X is a DGA, simply by
employing the ordinary cup product. The theory HI∗ also addresses questions in type
II string theory related to the existence of massless D-branes arising in the course of a
Calabi-Yau conifold transition. These questions are answered by IH∗ for IIA theory, and
by HI∗ for IIB theory; see Chapter 3 of [Ba10]. Furthermore, given a spectrum E in the
sense of stable homotopy theory, one may form EI∗(X) = E∗(Ip¯X). This, then, yields
p¯
an approach to defining intersection versions of generalized cohomology theories such as
K-theory.
Definition 2.1. The category CW of k-boundary-split CW-complexes consists of the
k⊃∂
following objects andmorphisms: Objects arepairs (K,Y), where K is a simply connected
CW-complex and Y C (K;Z) is a subgroup of the k-th cellular chain group of K that
k
⊂
arises as the image Y = s(im∂) of some splitting s : im∂ C (K;Z) of the boundary
k
→
map ∂ : C (K;Z) im∂( C (K;Z)). (Given K, such a splitting always exists, since
k k−1
→ ⊂
im∂ is free abelian.) A morphism (K,Y ) (L,Y ) is a cellular map f : K L such
K L
→ →
that f (Y ) Y .
∗ K L
⊂
6 MARKUSBANAGL ANDLAURENTIUMAXIM
Let HoCW denote the category whose objects are CW-complexes and whose mor-
k−1
phisms are rel (k 1)-skeleton homotopy classes of cellular maps. Let
−
t : CW HoCW
<∞ k⊃∂ k−1
−→
be the natural projection functor, that is, t (K,Y ) = K for an object (K,Y ) in
<∞ K K
CW , and t (f) = [f] for a morphism f : (K,Y ) (L,Y ) in CW . The
k⊃∂ <∞ K L k⊃∂
→
following theorem is proved in [Ba10].
Theorem 2.2. Let k 3 be an integer. There is a covariant assignmentt : CW
<k k⊃∂
≥ −→
HoCW of objects and morphisms together with a natural transformation emb : t
k−1 k <k
→
t such that for an object (K,Y) of CW , one has H (t (K,Y);Z) = 0 for r k,
<∞ k⊃∂ r <k
≥
and
emb (K,Y) : H (t (K,Y);Z) ∼= H (K;Z)
k ∗ r <k r
−→
is an isomorphism for r < k.
This means in particular that given a morphism f, one has squares
embk(K,YK)
t (K,Y ) // t (K,Y )
<k K <∞ K
t<k(f) t<∞(f)
(cid:15)(cid:15) embk(L,YL) (cid:15)(cid:15)
t (L,Y ) // t (L,Y )
<k L <∞ L
that commute in HoCW . If k 2 (and the CW-complexes are simply connected),
k−1
≤
then it is of course a trivial matter to construct such truncations.
Let p¯bea perversity. Let X beann-dimensional compact oriented pseudomanifold with
isolated singularities x ,...,x , w 1. We assume the complement of the singularities to
1 w
≥
be a smooth manifold. Furthermore, to be able to apply the general spatial truncation
Theorem 2.2, we require the links L = Link(x ) to be simply connected. This assumption
i i
is not always necessary, as in many non-simply connected situations, ad hoc truncation
constructions can be used. The L are closed smooth manifolds and a small neighborhood
i
of x is homeomorphic to the open cone on L . Every link L , i = 1,...,w, can be given
i i i
the structure of a CW-complex. If k = n 1 p¯(n) 3, we can and do fix completions
− − ≥
(L ,Y )ofL so thatevery (L ,Y )is anobject inCW . If k 2,nogroupsY have tobe
i i i i i k⊃∂ i
≤
chosen. Applying the truncation t : CW HoCW , we obtain a CW-complex
<k k⊃∂ k−1
→
t (L ,Y ) ObHoCW . The natural transformation emb : t t of Theorem
<k i i k−1 k <k <∞
∈ →
2.2 gives homotopy classes emb (L ,Y ) represented by maps
k i i
f : t (L ,Y ) L
i <k i i i
−→
such that for r < k,
f : H (t (L ,Y )) = H (L ),
i∗ r <k i i ∼ r i
while H (t (L ,Y )) = 0 for r k. Let M be the compact manifold with boundary
r <k i i
≥
obtained by removing from X open cone neighborhoods of the singularities x ,...,x .
1 w
SINGULARITIES AND INTERSECTION SPACE HOMOLOGY 7
The boundary is the disjoint union of the links,
w
∂M = L .
i
i=1
G
Let
w
L = t (L ,Y )
<k <k i i
i=1
G
and define a map
g : L M
<k
−→
by composing
f
L ∂M M,
<k
−→ −→
where f = f . The intersection space is the homotopy cofiber of g:
i i
DefinitionF2.3. The perversity p¯ intersection space Ip¯X of X is defined to be
Ip¯X = cone(g) = M cone(L ).
g <k
∪
Thus, to form the intersection space, we attach the cone on a suitable spatial homology
truncation of the link to the exterior of the singularity along the boundary of the exterior.
The two extreme cases of this construction arise when k = 1 and when k is larger than
the dimension of the link. In the former case, assuming w = 1, t (L) is a point and thus
<1
Ip¯X is homotopy equivalent to the nonsingular top stratum of X. In the latter case no
actual truncation has to be performed, t (L) = L, emb (L) is the identity map and thus
<k k
Ip¯X = X (again assuming w = 1). If the singularities are not isolated, one attempts to do
fiberwisespatialhomologytruncationappliedtothelinkbundle. Suchfiberwisetruncation
may be obstructed, however. If p¯= m¯ is the lower middle perversity, then we shall briefly
write IX for Im¯X. We shall put HIp¯(X) = H (Ip¯X) and HI (X) = H (IX); similarly for
∗ ∗ ∗ ∗
cohomology. When X has only one singular point, there are canonical homotopy classes
of maps
M IV V
−→ −→
described in Section 2.6.2 of [Ba10]. The first class can be represented by the inclusion
M ֒ IV. A particular representative γ : IV V of the second class is described in the
→ →
proof of Proposition 5.1. If V has several isolated singular points, the target of the second
map has to be slightly modified by identifying all the singular points. If V is connected,
then this only changes the first homology. The intersection homology does not change
at all. Two perversities p¯ and q¯ are called complementary if p¯(s) + q¯(s) = s 2 for all
−
s = 2,3,.... The following result is established in loc. cit.
Theorem 2.4. (Generalized Poincar´e Duality.) Let p¯ and q¯ be complementary perversi-
ties. There is a nondegenerate intersection form
HIp¯(X) HIq¯ (X) Q
i ⊗ n−i −→
which is compatible with the intersection form on the exterior of the singularities.
e e
8 MARKUSBANAGL ANDLAURENTIUMAXIM
The following formulae for HIp¯(X) are available (recall k = n 1 p¯(n)):
∗ − −
e H (M), i > k
HIp¯(X) = i
i H (M,∂M), i < k.
( i
e
In the cutoff-degree k, we have a T-diagram with exact row and exact column:
0
(cid:15)(cid:15)
0 // ker(H (M) H (M,L)) // H (M) // IH (X) // 0
k k k k
→
(cid:15)(cid:15)
HI (X)
k
(cid:15)(cid:15)
im(H (M,L) H (L))
k k−1
→
(cid:15)(cid:15)
0
The cohomological version of this diagram is
0
(cid:15)(cid:15)
0 // ker(Hk(M,L) Hk(M)) // Hk(M,L) // IHk(X) // 0
→
(cid:15)(cid:15)
HIk(X)
(cid:15)(cid:15)
im(Hk(M) Hk(L))
→
(cid:15)(cid:15)
0
When X is a complex variety of complex dimension n and p¯ = m¯, then k = n. If,
moreover, n is even, it was shown in [Ba10][Sect.2.5] that the Witt elements (over the
rationals) corresponding to the intersection form on IX and, respectively, the Goresky-
MacPherson intersection pairing on the middle intersection homology group, coincide. In
particular, the signature σ(IX) of the intersection space equals the Goresky-MacPherson
intersection homology signature of X. For results comparing the Euler characteristics of
the two theories, see [Ba10][Cor.2.14] and Proposition 4.6 below.
SINGULARITIES AND INTERSECTION SPACE HOMOLOGY 9
3. Background on Hypersurface Singularities
Let f be a homogeneous polynomial in n+ 2 variables with complex coefficients such
that the complex projective hypersurface
V = V(f) = x Pn+1 f(x) = 0
{ ∈ | }
has one isolated singularity x . Locally, identifying x with the origin of Cn+1, the singu-
0 0
larity is described by a reduced analytic function germ
g : (Cn+1,0) (C,0).
−→
Let B Cn+1 be a closed ball of radius ǫ > 0 centered at the origin and let S be its
ǫ ǫ
⊂
boundary, a sphere of dimension 2n+1. Choose ǫ small enough so that
(1) the intersection V B is homeomorphic to the cone over the link L = V S = g =
ǫ 0 ǫ
∩ ∩ {
0 S of the singularity x , and
ǫ 0
}∩
(2) the Milnor map of g at radius ǫ,
g
: S L S1,
ǫ 0
g − −→
| |
is a (locally trivial) fibration.
The link L is an (n 2)-connected (2n 1)-dimensional submanifold of S . The fibers of
0 ǫ
− −
the Milnor map are open smooth manifolds of real dimension 2n. Let F be the closure in
0
S of the fiber of g/ g over 1 S1. Then F , the closed Milnor fiber of the singularity is
ǫ 0
| | ∈
a compact manifold with boundary ∂F = L , the link of x . Via the fibers of the Milnor
0 0 0
map as pages, S receives an open book decomposition with binding L .
ǫ 0
Let π : X S be a smooth deformation of V, where S is a small disc of radius, say,
→
r > 0 centered at the origin of C. The map π is assumed to be proper. The singular
variety V is the special fiber V = π−1(0) and the general fibers V = π−1(s), s S, s = 0,
s
∈ 6
are smooth projective n-dimensional hypersurfaces. The space X is a complex manifold of
dimension n+1. Given V as above, we shall show below that such a smooth deformation
π can always be constructed. Let B (x ) be a small closed ball in X about the singular
ǫ 0
point x such that
0
(1) B (x ) V can be identified with the cone on L ,
ǫ 0 0
∩
(2) F = B (x ) V can be identified with F .
ǫ 0 s 0
∩
(Note that this ball B (x ) is different from the ball B used above: the former is a ball
ǫ 0 ǫ
in X, while the latter is a ball in Pn+1.) Let B = intB (x ) and let M be the compact
ǫ 0 0
manifoldM = V B with boundary ∂M = L . For 0 < δ < r, set S = z S z < δ ,
0 0 0 δ
− { ∈ | | | }
S∗ = z S 0 < z < δ and N = π−1(S ) B. Choose δ ǫ so small that
δ { ∈ | | | } δ δ − ≪
(1) π : N S is a proper smooth submersion and
δ δ
| →
(2) π−1(S ) N := N B.
δ δ
⊂ ∪
10 MARKUSBANAGL ANDLAURENTIUMAXIM
For s S∗, we shall construct the specialization map
∈ δ
r : V V.
s s
−→
By the Ehresmann fibration theorem, π : N S is a locally trivial fiber bundle projec-
δ δ
| →
tion. Since S is contractible, this is a trivial bundle, that is, there exists a diffeomorphism
δ
φ : N S M (recall that M is the fiber of π over 0) such that
δ δ 0 0
→ × |
φ
N // S M
δAπAA|AAAAA ∼=zzvvvvvvπvv1vvδ × 0
S
δ
commutes. The second factor projection π : S M M is a deformation retraction.
2 δ 0 0
× →
Hence ρ = π φ : N M is a homotopy equivalence. Let M be the compact manifold
δ 2 δ 0
→
M = V B with boundary L := ∂M, s S∗. We observe next that the composition
s − ∈ δ
M ֒ N ρδ M
δ 0
→ −→
is a diffeomorphism. Indeed,
M = π −1(s) = φ−1π−1(s) = φ−1( s M )
| 1 { }× 0
is mapped by φ diffeomorphically onto s M , which is then mapped by π diffeomor-
0 2
{ }×
phically onto M . This fixes a diffeomorphism
0
∼=
ψ : (M,L) (M ,L ).
0 0
−→
Thus L is merely a displaced copy of the link L of x and M is a displaced copy of the
0 0
exterior M of the singularity x . The restricted homeomorphism ψ : L L can be
0 0 0
| →
levelwise extended to a homeomorphism cone(ψ ) : coneL coneL ; the cone point is
0
| →
mapped to x . We obtain a commutative diagram
0
M oo L // cone(L)
ψ ∼= ψ| ∼= cone(ψ|) ∼=
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M oo L // cone(L ),
0 0 0
where the horizontal arrows are all inclusions as boundaries. Let us denote by W :=
M coneL the pushout of the top row. Since V is topologically the pushout of the
L
∪
bottom row, ψ induces a homeomorphism
∼=
W V.
−→
We think of W as a displaced copy of V, and shall work primarily with this topological
model of V. We proceed with the construction of the specialization map. Using a collar,
wemaywriteM asM = M [ 1,0] L withM acompact codimension0submanifold
0 0 0 0 0
∪ − ×
of M , which is diffeomorphic to M . The boundary of M corresponds to 1 L . Our
0 0 0 0
{− }×