Framed codimension 2 surgery
The algebraic results on asymmetric L-theory of Chap. 26 will now be applied to the codimension 2 surgery theory of Chap. 22 with trivial normal bundle ξ = є 2. This will be used describe the bordism of automorphisms of manifolds in Chap. 28, the algebrai
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The algebraic results on asymmetric £-theory of Chap. 26 will now be applied to the codimension 2 surgery theory of Chap. 22 with trivial normal bundle = E2 • This will be used describe the bordism of automorphisms of manifolds in Chap. 28, the algebraic theory of open books in Chap. 29, and highdimensional knot cobordism in Chap. 33.
e
27A. Codimension 1 Seifert surfaces In general, a knot k : Nn c Mn+2 need not be spanned by a codimension 1 Seifert surface. The following condition ensures the existence of such a spanning surface. Definition 27.1 Let (M, 8M) be ann-dimensional manifold with boundary. A codimension 2 submanifold (N, 8N) C (M, 8M) is homology framed if it is E2 -characteristic in the sense of 22.2, that is if [N]
=
0 E Hn-2(M,8M)
and there is given a particular identification VNcM
=
E2
:
N ~ BS0(2)
with an extension of the projection S(vNcM) projection on the exterior
= NxS 1 ~S 1 to the canonical
corresponding to a lift of
[N]
E ker(Hn-2(N, 8N)~Hn-2(M, 8M))
=
im(Hn-1 (M, N
to an element
A. Ranicki, High-dimensional Knot Theory © Springer-Verlag Berlin Heidelberg 1998
X
D 2 U 8+P)~Hn-2(N, 8N))
304
27. Framed codimension 2 surgery
Remark 27.2 The homology framing condition [N] = 0 E Hn-2(M, 8M) is equivalent to the condition VNcM = f. 2 if H 2 (M)----+H 2 (N) is injective. However, in general the two conditions are not equivalent. For example, N = {pt.} c M = S 2
has VNcM = E2 , but it is not homology framed (and does not admit a Seifert surface) - in fact, N C M is 17-characteristic, with 17 : M ----+BS0(2) the Hopf bundle, such that [N]
=
e(17)
=
1 E Ho(M)
=
H 2 (M)
=
0
Z.
Definition 27.3 Let (M, 8M) be ann-dimensional manifold with boundary, and let (N, 8N) c (M, 8M) be a homology framed codimension 2 submanifold with exterior (P, 8+P), so that
= N X D 2 UNxSl P , P = 8M = 8N X D 2 UaNxSl 8+P , M
cl.(M\N 8+P
=
X
D2 )
,
cl.(8M\8N x D 2 )
.
M
8M
(i) The canonical infinite cyclic cover of the exterior (P, 8+P) is the infinite cyclic cover obtained from the universal cover lR of 8 1 by pullback along the canonical projection (p, 8+p) : (P, 8+P)----+S 1 (P,8+P) = (p,8+p)*IR,
which restricts to (N, a+N) X lR over (N, 8N) X 8 1 ~ (P, a+P). (ii) A codimension 1 Seifert surface (F, 8+F) for (N, 8N) is a codimension 1 submanifold F C P such that 8F
=
8(8+F)
[F]
=
=
N UaN 8+F with 8+F
= 8N,
VFcP
= f.
:
F
n 8+P ,
F----+ B0(1) ,
p E Hn-l(M,N x D 2 U 8+P),
with 8+F a codimension 1 Seifert surface for 8N
c
8M.
27 A. Codimension 1 Seifert surfaces
305
M (iii) A Seifert fundamental domain for the canonical infinite cyclic cover P of P is the relative cobordism (PF; i 0 F, i 1 F) obtained by cutting P along a codimension 1 Seifert surface F C P, with ik : F - t F x {0, 1} c PF ; x - t (x, k) (k = 0, 1) ,
P
=
PF
UFx{O,l}
F
X
I .
The relative cobordism (o+Pa+Fi ioo+F, i1o+F) is then a Seifert fundamental domain for the canonical infinite cyclic cover o+P of o+P. D
sn-
sn
2 c M = Remark 27.4 The special case N = of 27.3 for n = 3 is the cla
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