Geometric TopologyHarvard University — Fall 2003 — Math 99rCourse NotesContents1 Introduction: Knots and Reidemeister moves . . . . . . . . . . . 12 1-Dimensional Topology . . . . . . . . . . . . . . . . . . . . . . . 13 2-Dimensional Topology . . . . . . . . . . . . . . . . . . . . . . . 34 Knot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Introduction: Knots and Reidemeister movesSimplicial complexes and embeddings. Any n-complex e mbeds in R2n+1.The utility graph and the pentagram do not embed in the plane.Knots and links. Fundamental group, the Hopf link, the unlink, and t hecarabiner trick.Knots exist. General position, Reidemeister moves, tricoloring, existence ofknots.A knot in the Science Center. Two people A and B walk from up the mainstairs from floor 1 to floor 3 – arriving outside 309 – holding hands. They startwith their backs to us as seen from the yard, in position A-B. The arrive onthe 3rd flo or facing us, so in postion B-A. Now B goes down one wing of theSC – toward Littauer – and A goes down the other wing – t o the new Hist. ofSci. dept. They go down the stairs in their wings, arrive on the first floor, andrejoin. Now they are in postion B-A with their backs to us. So the two pathsthey have traced join toge ther to form a closed path = a knot. The knot issimply the closure of a braid on two strands with an odd number of twists. Thenumber of twists comes out to be 3 – which gives the trefoil.2 1-Dimensional TopologyGroups. Free group, presentations, Tietze moves, unsolvable problems intopology. Free products and semidirect products; the word problem for each.The groups ha, b : a2, b2i and ha, b : a2, aba′= b′i; the dihedral groupsha, b : a2, aba′= b′, bni.PSL2(Z) = ha2, b3i and the groups Gn= ha2, b3, (ab)ni; G2,3,4,5= S3, A4, S4, A5.Cayley graphs. The fundamental group and universal cover of X = S1∧ S1.Deck groups and irregular coverings. The coverings of X coming from S3andits subgroups.Nielsen-Schreier Theorem: any f.g. subgroup of a free group is free. For anyfinite graph, χ( X) = 1 − n where π1(X)∼=Fn. If X → Y is a degree d covering1of graphs, then χ(X) = dχ(Y ). Thus if H ⊂ Fnhas index d, then H∼=Fmwith1 − m = d(1 − n).Univer sal covers. The universal cover of S1. The degree of a map f : S1→S1. Classification of immersions; smiles and frowns.The universal cover of S2, RP2and S1× S1. The product formula: π1(A ×B) = π1(A) × π1(B).More covering spaces of the bouquet of 2 circles X:H = hai, ha, b2i, h bnab−n: n ∈ Zi.The kernel of Z ∗ Z → Z is infinitely generat e d. Generators and normal gener-ators for the kernel of Z ∗ Z → S3.Amenability. A graph is amenable if inf |∂V |/|V | = 0; a group, if its Cayleygraph is amenable. Ponzi schemes and the degree three tree; the enrichmentflow shows |∂V | ≥ |V |.The group Znis amenable, the fre e group is not. A non-amenable group hasexponential growth, but not conversely.An ame nable group of exponential growth. Let G = ha, b : ab = b2ai. Itis easy to see that G has exponential growth, and in fact G is isomorphic to thegroup of affine transformations of R of the form f(x) = 2nx + k2m, n, m, k ∈ Z.Here is a sketch of a proof that G is amenable. Consider the elements of Gof the form g(i, j) corresponding to the transformations 2−i(x + j), with i ≥ 1.Then we haveag(i, j) = g(i − 1, j) and bg(i, j) = g(i, j + 2i).This makes it easy to visualize a large part of the Cayley graph of G. For aneven better picture, mod out by the action of b (look at the hbi-cosets): theresult is a collection of ‘circles’ Si∼=Z/2i, with the a-edges joining the pointsof Sito Si−1in pairs. The result is a bifurcating tree T .Thus we can think of the Cayley graph as a tree T , with each vertex replacedby a copy of Z. The a-edges converge in pairs, but this is compensated for bythe fact t hat the b-edges spread apart, interleaving the copies of Z.Now consider the set V = {g(i, j) : 1 ≤ i ≤ L, 1 ≤ j ≤ M}. Then using theequations above, it is easy to see that the b-edges contribute about 2ivert ice s to∂V at level i, resulting in a total of about 2L. The a-edges obviously contributea boundary of size M . Thus we have|∂V ||V |≍2L+ MLM·By taking 0 ≪ L ≪ 2L≪ M, we can make this ratio arbitrarily small, andtherefore G is amenable.General theory of the fundamental group. Homotopy equivalence andretracts. Contracting trees in graphs (1) to show homotopy equivalence to abouquet of circles and (2) to show covering spaces are homotopy equivalent tofinite graphs.2Theta and barbells. To see these are homotopy equivalent, note they areboth deformation retracts of a pair of pants.Collapsing: a c ompact surface Σ with nonempty boundary is homotopyequivalent to a graph. Thus π(Σ) is free.Collapsing: the house wit h two rooms. Not every contractible complexcollapses.Seifert-van Kampen. A present ations for G ∗AH is given by the union ofthe presentations for G and H, together with the additional relations φG(ai) =φH(ai) for each generator of A. (Note that the relators of A are irrelevant.) I fH is trivial, then G ∗AH = G/N(φG(A)).The Seifert-van Kampen theorem saysπ1(X ∪ Y ) = π1(X) ∗π1(X∩Y )π1(Y ).Examples: bouquets of circles; π1(Sn); π1(S1× S1), thought of as a toruswith a hole, with the hole glued back in.Presentations and cell complexes. A compact Hausdorff space X is a finitecell complex if there are maps fi: Dni→ X, whose image s cover X, such thatfiis a homeomorphism on the interior of the ni-disk, and fisends the boundaryinto the (ni− 1) skeleton of X, defined as the union of the cells of dimensionk ≤ ni− 1.Theorem: given any finitely presented group G, there is a finite 2-complexK such that π1(K)∼=G.Example: The two complex canonically associated to Z2= ha, b : [a, b]i is atorus.Theorem (Poincar´e): for any complex K, we have π1(K) = π1(K(2)).First betti number. The cohomology group H1(G, Z) = G/[G, G]. Example:H1(Fn, Z)∼=Zn. This shows free groups on different numbers of generators arenot isomorphic.For a topological space X, we define b1(X) = rank of free part of H1(G, Z).Also b0(X) = number of components of X. Then for a graph we have:χ(X) = b0(X) − b1(X).This generalizes, and shows χ(X) is a homotopy invariant.3 2-Dimensional TopologyBackground. Closed manifolds, manifolds with boundary. A combinatorialn-manifold is a simplicial complex such that the link of every vertex is a