Math 872 - Algebraic Topology - Spring 2010

(Note: This page is a shameless rip-off from a similar page for Susan Hermiller's Spring 2009 version of the same course.)

* indicates the location of the class currently.

Chapter 0: Overview of the course

• Section A: Big questions in topology

• Classification problem

• Homeomorphism problem and homotopy equivalence problem

• Section B: Homeomorphism invariants

• Groups - homotopy groups

• Abelian groups - homology groups

• "Categories" and "functors"

Chapter 1: Review: Quotients, retracts, and homotopy

• Section A: Quotient spaces

• Identification space and topology

• Thm: If p: X -> X/∼ is a quotient map and g: X -> Z is continuous and constant on equivalence classes, then g induces a continuous f: X/∼ -> Z with f ο p = g.

• Cor *: If g:X -> Y is a continuous surjection, x1 ∼ x2 iff g(x1)=g(x2) for all x1,x2 ∈ X, X is compact, and Y is Hausdorff, then X/∼ ≅ Y.

• Disjoint union topology

• Mapping cylinder, examples

• Section B: Retracts

• Definition of retract (space) and retraction (map)

• Definition of deformation retract (space) and deformation retraction (homotopy/map)

• Thm: For the mapping cylinder Xf of the map f: X -> Y, Y is a deformation retract of X.

• Examples

• Section C: Homotopy

• Definitions of homotopy and homotopy relative to a subspace

• Definitions of homotopy equivalence (maps) and homotopy type (equiv class)

• Lemma: Homotopy is an equivalence relation on maps, homotopy equivalence is an equivalence relation on spaces.

• Thm: If Y is a deformation retract of X, then they have the same homotopy type.

• Thm: If X and Y have the same homotopy type, then there is a space Z that deformation retracts to both X and Y.

• Contractible space, nullhomotopic map

• Examples

Chapter 2: Fundamental groups

• Section A: Definition of π1

• Loop, path homotopy, basepoint

• Product of loops, constant loop, reverse of a loop

• Def: π1(X)

• Thm: π1 is a group

• Examples: Rn and the straight-line homotopy, S2

• Change of basepoint map induced by a path

• Thm: The change of basepoint map is an isomorphism of groups.

• Section B: Induced homomorphisms

• Group homomorphism induced by a map of pointed spaces

• Lemmata: Maps induced by continuous functions are group homomorphisms that compose nicely and commute with maps induced by paths.

• Thm: If X and Y are path-connected and have the same homotopy type, then π1(X) ≅ π1(Y).

• Section C: π1(S1)

• Examples of spaces with trivial fundamental group

• Thm: The fundamental group of a contractible space is trivial.

• Path Lifting Theorem and Homotopy Lifting Theorem for lifting paths and homotopies from S1 to R1

• Thm: π1(S1) ≅ Z

• Proof of the Path Lifting Theorem

• Proof of the Homotopy Lifting Theorem

• Borsuk-Ulam Theorem: proof and consequences.

• Thm: π1(X × Y) is isomorphic to π1(X) × π1(Y).

Chapter 3: Presenting groups

• Section A: Review and Presentations

• Review of normal subgroups, cosets, and quotients

• Review of the Isomorphism Theorems

• Definitions and examples of free groups

• Definitions and examples of presentations of groups *

• Homomorphism Building Theorems

• Tietze transformations and Tietze's Theorem

• Section B: Building new groups from old

• Three views of direct products: Sets with multiplication, presentations, and the universal property

• Three views of free products, and examples

• Amalgamated products

Chapter 4: The Seifert - Van Kampen Theorem

• Section A: Statement and first examples

• SVK theorem statement

• Computing fundamental groups of graphs and surfaces

• Abelianization of a group

• Applications to the homotopy equivalence problem

• Finitely presented group examples and the 2-way street

• Section B: Proof of SVK

• Building the homomorphism with the UP

• Using the Lebesgue Number Lemma to prove onto

• Using LNL again to prove 1-1

Chapter 5: Presenting spaces

• Section A: Building new spaces from old

• Wedge products

• Thm: If for each α the basepoint xα of each space Xα is a deformation retract of an open neighborhood Uα of xα in Xα, then π1(∨α Xα) ≅ ∗α π1(Xα) is a free product.

• Cones

• Section B: CW complexes

• Definitions: cells, n-skeleton, characteristic map, dimension

• Examples, including putting CW structures on familiar spaces

• Thm: A finite CW complex is compact.

• Thm: CW complexes are Hausdorff.

• Section C: Fundamental groups of CW complexes

• Thm 1: The fundamental group of the 1-skeleton of a CW complex is a free group, generated by loops at a basepoint that follow a path in a maximal tree T, then traverse a single edge outside T, and then another path in T back to the basepoint.

• Thm 2: For a CW complex X the inclusion X(1) -> X(2) induces a surjection of fundamental groups whose kernel is generated by loops corresponding to the attaching maps of the 2-cells of X.

• Thm 3: π1(X) is isomorphic to π1(X(2)).

• Presentation complexes

• 2-Way Street Thm: For every group G there is a 2-dimensional CW complex X with π1(X) isomorphic to G.

Chapter 6: Covering spaces

• Section A: Connectednesses

• PC and LPC, and examples

• Simply-connected and SLSC, and examples

• Thm: CW complexes are LPC and SLSC.

• Section B: Definitions and lifting

• Definition and examples

• Path, path homotopy, PPHLT, and homotopy lifting theorems

• Thm: ker p* = 1 and im p* = {[f]| f lifts to a loop}.

• Thm: For all natural numbers n, the free group Fn is a subgroup of the free group F2.

• Lifting Criterion and Unique Lifting Property, and examples

• Lifting Correspondence Function and Thm: Let X and Y be path-connected spaces and let p: (Y,y0) -> (X,x0) be a covering space. The Lifting Correspondence Function Φ: π1(X,x0) / p*1(Y,y0)) -> p-1({x0}) defined by Φ(p*1(Y,y0))[l]) := m(1), where m is the unique lift of the path l in Y starting at y0, is a well-defined bijection.

• Thm: If p: Y -> X is a covering space and X is a CW complex, then Y is a CW complex.

• Section C: Group actions

• Actions, covering space actions, orbits, and orbit spaces

• Thm: If G has a covering space action on Y, then: (1) The quotient p:Y -> Y/G is a covering space. (2) If Y is PC and LPC then p*1(Y,y0)) is normal in π1(Y/G,[y0]). (3) If Y is PC and LPC then G is isomorphic to π1(Y/G,[y0]) / p*1(Y,y0)).

• Cor: If Y is simply-connected and LPC and G has a covering space action on Y, then G is isomorphic to π1(Y/G).

• Definitions of presentation complex, Cayley graph, and Cayley complex, and examples.

• Thm: Let Y be the Cayley complex of a presentation of G. Then G has a covering space action on Y, Y/G is the presentation complex, Y is simply connected, and π1(Y/G) ≅ G.

• Section D: The universal covering and the Galois correspondence

• Simply-connected Covering Thm: Let X be a PC, LPC, SLSC space. Then there is a simply-connected covering space p: Y -> X, and there is a covering space group action of π1(X) on Y inducing the map p.

• Cor.: Let X be a PC, LPC, SLSC space, and let H be a subgroup of π1(X). Then there is a covering space p: Y -> X with H=p*1(Y)), and there is a covering space group action of H on Y inducing the map p.

• The Galois correspondence and examples

• Thm: Every subgroup of a free group is free.

• Definition of isomorphism of (pointed) covering spaces

• Thm: Any two PC, LPC pointed covering spaces pi: (Xi,xi) -> (X,x0) (i=1,2) of a pointed space (X,x0) satisfy p1*(X1,x1) = p2*(X2,x2) iff the pointed covering spaces are isomorphic.

• Galois Correspondence Thm: Let (X,x0) be a PC, LPC, SLSC space, and let p: (Y,y0) -> (X,x0) be the simply-connected covering. Then the maps {subgroups H of G} <-> {isomorphism classes of pointed coverings of (X,x0)} defined by H -> (p': (Y/H,[y0]) -> (X,x0)) and (p'': (Y'',y''0) -> (X,x0)) -> im p''* are inverse bijections.

• Universal covering thm: If X is PC, LPC, and SLSC, p: Y -> X is the unique simply-connected covering space, and q: Z -> X is another covering space, then there is a covering r: Y -> Z with the composition qr=p.

• Deck transformations and examples

• Thm: Let p:(Y,y0) -> (X,x0) be a covering space with Y PC and X PC, LPC, and SLSC. Then Deckgp(p) ≅ N(H)/H, where H = p*1(Y,y0)).

• Examples and normalizers of subgroups

• Normal covering spaces

• Thm: Let p:(Y,y0) -> (X,x0) be a covering space with Y PC and X PC, LPC, and SLSC. TFAE: i) The covering space is normal. ii) Deckgp(p) has a covering space action on Y inducing p. iii) p*1(Y,y0)) is a normal subgroup of π1(X,x0).

Chapter 7: Simplicial homology

• Section A: Overview of homology

• Higher homotopy groups and their weaknesses

• Strengths and weaknesses of simplicial, CW, and singular homologies

• Section B: Δ-complexes

• Simplices and faces

• Δ-complex definition and examples

• Section C: Simplicial chains

• Simplicial n-chains and boundary maps ∂n

• Examples

• Connections between ker ∂1 and loops, im ∂2 and disks filling in loops

• Lemma: ∂n-1 ο ∂n

• Section D: Computing simplicial homology

• Definition of Hnsimpl

• Examples

• Reduced simplicial homology

• Thm: If a topological space has two Δ-complex structures, i.e. if X and X' are homeomorphic Δ-complexes, then Hnsimpl(X) is isomorphic to Hnsimpl(X') for all n.

Chapter 8: Singular homology

• Section A: Definitions and examples

• Singular chain complexes and definition of Hnsing

• Thm: If X has path components Xα, then Hnsing(X) = ⊕α Hnsing(Xα).

• Thm: If X is path-connected, then H0sing(X) = Z.

• Thm: If X is a Δ-complex, then Hnsimpl(X) and Hnsing(X) are isomorphic for all n.

• Homological algebra: chain complexes, cycles, boundaries, homology, chain maps, homology homomorphisms, chain homotopy, and chain homotopy equivalence

• Inducing singular homology homomorphisms via continuous functions

• Lemmata: Homology homomorphisms induced by continuous functions are abelian group homomorphisms that compose nicely; homotopic maps induce the same homology homomorphism.

• Thm: If X and Y are homotopy equivalent, then Hnsing(X) and Hnsing(Y) are isomorphic for all n.

• Examples/applications

• Section B: Mayer-Vietoris Theorem

• Connection to the SVK theorem

• Small chains thm: If A,B are subspaces of X with X=int(A) ∪ int(B) then Hn(X) is isomorphic to Hn(A+B).

• Homological algebra: A short exact sequence of chain complexes induces a long exact sequence on homology.

• MV theorem statement, and proof using the small chains thm

• Examples of computing with the Mayer-Vietoris Theorem

C Bleak. Last update 2/12/2010