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206

APPENDIX A GLOSSARY FLOW CHARTS

& theory

injective

union surjective quotient map intersection bijective

inverse map

set

element equivalence relation

pre-image

object

map

category 207

point-set

compactification complete quotient space Hausdor↵ imbedding metric completion gluing paracompact isometry cutting

interior continuous cauchy subspace topology compact metric topology convergent point connected limit

topology

open metric

closed

&

set theory 208

numbers and algebra

modular arithmetic vector space real numbers, R action Z/nZ

group integers, Z rational numbers, Q field

binary operation

identity element point-set topology

cardinality inverse operation metric completion natural numbers, Z+ commutative cauchy, convergence

associative metric, sequence

linear

category theory

&

set theory 209

linear algebra

linear

determinants

Cramer’s rule

multilinear maps

tensor products

exterior products

biinear maps

symmetric

skew-symmetric

definite

the plane, R2 linear maps 3-space, R3 bases ordered bases rank inner products n space, Rn orientation nullity

numbers and algebra

real numbers, R

module, vector space

ring, field 210

differential vector calculus

critical values the inverse function theorem the regular value theorem the implicit function theorem the constant rank level set theorem

higher order mixed partial derivatives

Rm n smooth maps A Jacobian in ⇥ represents the m n m immersions and submersions total derivative Dfa of f : R R at a R . ! 2 di↵eomorphisms and embeddings

partial derivatives @ @xi a m n m of f : R R at a R ! 2

linear algebra

linear maps f : Rn Rm point-set topology ! multilinear map, exterior product continuity, limit

linear isomorphisms, determinants 211

transition functions and , submanifolds oriented manifolds S0,S1,S2,S3,Sn,and knots and links in S3 smooth manifolds projective spaces, RPn spanning surfaces in S3 Riemannian manifolds

topological manifolds

(with or without boundary)

coordinate charts

atlases of charts

linear algebra point-set topology n-space,Rn differential vector calculus continuity, quotient space multilinear map, exterior product higher order mixed partial derivatives cutting, gluing linear isomorphisms, determinants smooth maps Hausdor↵, paracompact inner products Jacobian matrices compact, connected symmetric bilinear mappings interior, boundary, closure orientation 212

covering spaces and fundamental groups

structure theorem

universal covering spaces the path connectedness concatenation of paths local path connectedness the lifting correspondence semilocal simply-connectedness

paths covering spaces homotopy of paths example: R S1 ! homotopy of maps

point-set topology

continuity, quotient space algebra cutting, gluing groups Hausdor↵, paracompact group actions compact, connected

interior, boundary, closure 213

APPENDIX B HOMOLOGY AND KNOTS

B.1 Overview

Working in the smooth (or piecewise linear) category ensures that every link L S3 ⇢ is tame, meaning that its embedding : n S1 , L S3 extends to an embedding r=1 ! ⇢ of disjoint, compact, solid tori : n SF1 D2 , S3, where (x,0)= (x) for each r=1 ⇥ ! ⇣ ⌘ x n S1. Call the image of a closedF regular neighborhood, or thickening of L, and 2 r=1 F denote this image by ⌫L. Also let ⌫Ls and ⌫Ls respectively denote the closed and open

1 regular neighborhoods of any component Ls = r=s S of L, and let s, s denote the

⇣ ⌘ 3 restrictions = 1 , = . The openF 3- S L is called the link s r=s S s (S1 D2) | | r=s ⇥ \ F complement; the compact 3-manifoldF with boundary S3 ⌫L is called the link exterior. \ Link exteriors provide an (almost too) intuitive setting in which to introduce the basic machinery of ; by contrast, the traditional approach to the sub- ject traces its historical development, following the axiomatic constructions of and singular homology, proving their equivalence, and then deriving proper- ties. This traditional approach is necessary in order to build a solid, logical foundation for homology theory; flowcharts in the Appendices reflect these logical foundations. For ex- pository purposes, however, homology theory for 3-manifolds is all about submanifolds.

If M is a 3-manifold, then we consider the following eight homology groups:

Z F Hk(M; ) and Hk(M; 2) for k =0,1,2,3, taking coecients in the integers or in the two-

F Z Z Z F element field 2 = /2 . Each element of each Hk(M; ) (resp. Hk(M; 2)) can be seen 214

Figure B.1: Left: a longitude and meridian µ on the boundary @⌫L of the thickened link ⌫L, in case L is the lh trefoil. Right: a Seifert F S3 ⌫L with @F = . ⇢ \

as an equivalence class [s] of oriented (resp. unoriented) embedded submanifolds s M ⇢ of dimension n (this fails when M has higher dimension). The equivalence is straightfor- ward in the case of disjoint, connected submanifolds: taking coecients in the integers

Z, a disjoint pair of connected, n-dimensional submanifolds s ,s M are equivalent, 1 2 ⇢ [s ]=[s ], i↵ s s bounds an oriented (n+1)-dimensional submanifold S M, such that 1 2 1 [ 2 ⇢ the induced orientation on @S = s s preserves the orientation on s and reverses that 1 [ 2 1 on s2. In particular, [s1] = 0 and s1 is called nullhomologous if s1 bounds an oriented submanifold S M. The equivalence works the same with coecients in Z/2Z, omitting ⇢ everything about orientation.

Let M = S3 ⌫L be a link exterior, and let M = S3-cut-along-F be a link exterior 1 \ 2 Z Z cut along a spanning surface. In the case of such Mi, i =1,2, both H0(Mi; )= and

F F Z F H0(Mi; 2)= 2, since both Mi are path connected, and both H3(Mi; )=0=Hx(Mi, 2), since both Mi, although compact and oriented, have nonempty boundary. Each primitive 215

Z F element of H1(Mi; ) (resp. H1(Mi; 2)) can be seen as a simple closed curve that bounds

Z F no oriented (resp. unoriented) surface in Mi, and elements of H2(Mi; ) and H2(Mi; 2) can be seen as embedded surfaces F M (compact and without boundary) that do not ⇢ i bound.

The first homology groups, H1, will be the most useful of these groups, because of the linking pairing lk : H (X;Z) H (Y;Z) Z for any disjoint subsets X,Y S3, which 1 ⌦ 1 ! ⇢ is determined by the following property. If ↵ = f (S1) X, = g(S1) Y are simple closed ⇢ ⇢ curves, then their lk(↵,) is the degree of a map h defined as follows. Let z S3 (X Y) be any point, let ' : S3 z R3 be stereographic projection, and let 2 \ [ \{ } ! h : S1 S1 S2 be given by ⇥ !

' g(y) ' f (x) h :(x,y) . (B.1) 7! ' g(y) ' f (x) 3 The degree of this map can be seen, by choosing an “upward” direction in R , as the number of times ↵ passes directly above in a right-handed way, minus the number of times ↵ passes directly above in a left-handed way:

lk(↵,)= β β . (B.2) α α This gives:

L L Lr lk(L ,L )= r s L . (B.3) r s s B.2 Longitudes, meridia, and boundary slopes

Given a thickening ⌫L of a link L, first homology groups coming from each link component Ls distinguish two isotopy classes of curves on @⌫Ls, the longitude s and 216

the meridian µs, which in turn give rise to the extremely useful notion of boundary

slopes. A meridian µs is any simple closed curve on @⌫Ls that represents a generator

of H (S3 ⌫L ;Z) and is nullhomologous in H (⌫L ;Z). For example, p @D2 is 1 \ s 1 s r=s{ } ⇥ ⇣ ⌘ a meridian for any point p S1.Alongitude is any simple closed curveF on @⌫L that 2 s s is nullhomologous in H (S3 ⌫L ;Z) and is a generator of H (⌫L ;Z). Thus, µ bounds 1 \ s 1 s s

an oriented surface (in fact a ) in ⌫Ls but not in @⌫Ls, and a longitude s bounds an

oriented surface in S3 ⌫L but not in @⌫L (c.f. Figure B.1). The Mayer-Vietoris sequence \ s s 3 for S -cut-along @⌫Ls proves that any two meridia on @⌫Ls are isotopic in @⌫Ls, as are any

two longitudes (c.f. the Appendix).

Recall, given a spanning surface F for a link L, that we isotope F to intersect each

component ⌫L of the thickening ⌫L in an annulus. Also assume that each pair (L ,F s s \ @⌫L ) is co-oriented, i.e. that these curves are oriented so that [L ]=[F @⌫L ] in H (⌫L ). s s \ s 1 s The sum of linking numbers n lk(L ,F @⌫L ) is called the slope of F, and the n- s=1 s \ s tuple (lk(L ,F @⌫L ))n is calledP the boundary slope of F. The latter characterizes the s \ s s=1 isotopy class of F @⌫L on @⌫L: each F @⌫L is homologous on @⌫L to [ ] + lk(L ,F \ \ s s s s \

@⌫Ls)[µs] when µs is oriented so that lk(Ls,µs) = +1, and simple closed curves on the torus

are determined by their homology class.

3 1 3 Explicitly, take an embedding ⇠ : F˙ , S , with ⇠ ˙ = : S , S (and = ! |@F s ! th ˙ 1F ˙ s s, where each s is the restriction of to the s copy Ls of S , and each s(Ls)=Ls

isF the sth component of L), and let ' : S3 z R3 be stereographic projection as before, \{ } ! ˙ where z S3 ⇠(F). Also take L to be a pusho↵ of L˙ into F˙, such that the restriction 2 \ s s ˙ ˙ ˙ s := ⇠ ˙ maps Ls to F @⌫Ls =:bLs. Whether or not F is orientable, orient L, and co-orient |Ls \ b b b b 217

˙ ˙ L := s Ls. We already saw (c.f. (B.1), (B.2), (B.3)) that each linking number lk(Lr,Ls)

F ˙ ˙ 2 ' s(y) ' r (x) bequals theb degree of the map hrs : Lr Ls S given by hrs :(x,y) . ⇥ ! 7! ' s(y) ' r (x) | | th Similarly, the s component of the slope of F, denoted by slopes(F), equals the degree of

the map h : S1 S1 S2 which is given by s ⇥ !

' s(y) ' s(x) hs :(x,y) . 7! ' s(y) ' s(x) b R3 This map can also be seen by choosing anb “upward” direction in and counting the

number of times L passes directly above L in a right-handed way and in a left-handed

way. This leads to: b

L L Ls slope (F)= + s s L . (B.4) s s s s Therefore, the net slope of F equals

slope(F)= slopes(F) s X L L Ls = + s s L s s s s X (B.5) Lr Lr Ls = + Ls r

1 1 slope(F)+ lk(L ,L ) = w(D)+ (B.6) 2 r s 2 2 r