We work in the smooth category.
An (oriented) (ordered) m-component n-(dimensional)
link is a smooth oriented submanifold
L = {K1, …, Km} of
Sn+2 which is the ordered disjoint union of m
manifolds, each PL-homeomorphic to the standard n-sphere. If
m = 1, then L is called a knot.
We say that m-component n-dimensional links L0
and L1 are (link-)concordant
or (link-)cobordant if there is a smooth oriented submanifold
C˜ = {C1, …, Cm}
of Sn+2 × [0, 1]
which meets the boundary transversely in ∂C˜, is PL-homeomorphic
to L0 × [0, 1], and meets
Sn+2 × {l} in Ll
(l = 0, 1). If m = 1, then we say that
n-knots L0 and Ll are
(knot-)concordant or (knot-)cobordant.
Then we call C a concordance-cylinder of the two n-knots
L0 and Ll.
If an n-link L is concordant to the trivial link,
then we call L a slice link.
If an n-link L
= {K1, …, Km}
⊂ Sn+2
= ∂Bn+3 ⊂ Bn+3
is slice, then there is a disjoint union of (n + 1)-discs
Dn+11 [amalg ] … [amalg ]
Dn+1m in
Bn+3 such that
Dn+1i ∩ Sn+2
= ∂Dn+1i
= Ki. (Dn+11,
…, Dn+1m) is called
a set of slice discs for L. If m = 1, then
Dn+11
is called a slice disc for the knot L.