**Math 872 -
Algebraic Topology - Spring 2010 **

**(Active) Table of Contents **

(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, x

_{1}∼ x_{2}iff g(x1)=g(x_{2}) for all x_{1},x_{2}∈ 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 X

_{f}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 π*_{1}Loop, path homotopy, basepoint

Product of loops, constant loop, reverse of a loop

Def: π

_{1}(X)Thm: π

_{1}is a groupExamples:

**R**^{n}and the straight-line homotopy, S^{2}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 π

_{1}(X) ≅ π_{1}(Y).

*Section C: π*_{1}(S^{1})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 S

^{1}to**R**^{1}Thm: π

_{1}(S^{1}) ≅**Z**Proof of the Path Lifting Theorem

Proof of the Homotopy Lifting Theorem

Borsuk-Ulam Theorem: proof and consequences.

Thm: π

_{1}(X × Y) is isomorphic to π_{1}(X) × π_{1}(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 x

_{α}of each space X_{α}is a deformation retract of an open neighborhood U_{α}of x_{α}in X_{α}, then π_{1}(∨_{α}X_{α}) ≅ ∗_{α}π_{1}(X_{α}) is a free product.Cones

*Section B: CW complexes*Definitions: cells,

*n*-skeleton, characteristic map, dimensionExamples, 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: π

_{1}(X) is isomorphic to π_{1}(X^{(2)}).Presentation complexes

2-Way Street Thm: For every group G there is a 2-dimensional CW complex X with π

_{1}(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 F

_{n}is a subgroup of the free group F_{2}.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,y

_{0}) -> (X,x_{0}) be a covering space. The Lifting Correspondence Function Φ: π_{1}(X,x_{0}) / p_{*}(π_{1}(Y,y_{0})) -> p^{-1}({x_{0}}) defined by Φ(p_{*}(π_{1}(Y,y_{0}))[l]) := m(1), where m is the unique lift of the path l in Y starting at y_{0}, 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

_{*}(π_{1}(Y,y_{0})) is normal in π_{1}(Y/G,[y_{0}]). (3) If Y is PC and LPC then G is isomorphic to π_{1}(Y/G,[y_{0}]) / p_{*}(π_{1}(Y,y_{0})).Cor: If Y is simply-connected and LPC and G has a covering space action on Y, then G is isomorphic to π

_{1}(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 π

_{1}(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 π

_{1}(X) on Y inducing the map p.Cor.: Let X be a PC, LPC, SLSC space, and let H be a subgroup of π

_{1}(X). Then there is a covering space p: Y -> X with H=p_{*}(π_{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 p

_{i}: (X_{i},x_{i}) -> (X,x_{0}) (i=1,2) of a pointed space (X,x_{0}) satisfy p_{1*}(X_{1},x_{1}) = p_{2*}(X_{2},x_{2}) iff the pointed covering spaces are isomorphic.Galois Correspondence Thm: Let (X,x

_{0}) be a PC, LPC, SLSC space, and let p: (Y,y_{0}) -> (X,x_{0}) be the simply-connected covering. Then the maps {subgroups H of G} <-> {isomorphism classes of pointed coverings of (X,x_{0})} defined by H -> (p': (Y/H,[y_{0}]) -> (X,x_{0})) and (p'': (Y'',y''_{0}) -> (X,x_{0})) -> 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,y

_{0}) -> (X,x_{0}) be a covering space with Y PC and X PC, LPC, and SLSC. Then Deckgp(p) ≅ N(H)/H, where H = p_{*}(π_{1}(Y,y_{0})).Examples and normalizers of subgroups

Normal covering spaces

Thm: Let p:(Y,y

_{0}) -> (X,x_{0}) 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_{*}(π_{1}(Y,y_{0})) is a normal subgroup of π_{1}(X,x_{0}).

**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 loopsLemma: ∂

_{n-1}ο ∂_{n}

*Section D: Computing simplicial homology*Definition of H

_{n}^{simpl}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 H

_{n}^{simpl}(X) is isomorphic to H_{n}^{simpl}(X') for all n.

**Chapter 8: Singular homology **

*Section A: Definitions and examples*Singular chain complexes and definition of H

_{n}^{sing}Thm: If X has path components X

_{α}, then H_{n}^{sing}(X) = ⊕_{α}H_{n}^{sing}(X_{α}).Thm: If X is path-connected, then H

_{0}^{sing}(X) =**Z**.Thm: If X is a Δ-complex, then H

_{n}^{simpl}(X) and H_{n}^{sing}(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 H

_{n}^{sing}(X) and H_{n}^{sing}(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 H

_{n}(X) is isomorphic to H_{n}(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* *