Algebraic Topology: Fall 2026
                            
                                Tuesday and Thursday 2:40 -- 3:55
                                        ROOM 507 Mathematics
                    
                                Instructor:  John Morgan,  603 Mathematics
                                                    jmorgan@math.columbia.edu

                                Office hours: Tuesday 4 pm - 5 pm

                                TA:       Peter Moody  ptm2131@columbia.edu
                                             Office Hours: Friday  10 - 11 am in 622 Math
                                               

                                Course Website:  www.math.columbia.edu/~jmorgan

                                FINAL EXAM: TBA
                                REVIEW SESSION:  TBA
                                     


 

                          

Prerequisites: I will assume basic knowledge in the following areas:

(i) basic terminology of category theory: categories, functors, natural transformations
(ii) linear algebra including matrices, tensor products and endomorphisms,
(iii) point-set topology,, including notions of Hausdorff, compact, locally compact
(iv) differentiable manifolds, including, inverse and implicit function theorems,
vector fields and their integral curves,       
(v) group theory, subgroups, cosets, normal sub-groups and quotient groups,
(vi) algebra including associative algebras (with unit), principal integral domains (PIDs).

  There are background notes introductory notes on these topics on the course website. 
 

Course Structure and Grades:

The only exam in the course will be an in-class final exam at the end on the semester.

Every week or two as the course proceeds, I will assign homework with a due date.
Homework will be posted on the website together with he due date. I will collect it in class.
The homework will be graded by the TA and returned to you.
The TA will also hold review sessions each week to go over the homework and answer
questions about the lectures. All homework solutions must be hand-written.

Grades will be assigned based performance on the homework and final exam
with the final having largest weight.

PROBLEM SETS:

Problem Set 1 is here and is due in class on Thursday, Sept. 24.
Problem Set 2 is here and is due in class on Thursday, Oct. 1.
Problem Set 3 is here and is due in class on Thursday, Oct. 8.


Reading Material:

Here, I have posted notes on background material that I will be assuming in class, I will post  notes of the lectures on the class website. Other references include Allen Hatcher's book and Edwin Spanier's book, both entitled "Algebraic Topology."
 
The Course is divided into 2 parts. 

I. Basics of Algebraic Topology
 (i) The homotopy category
 (ii) Definition of the homotopy groups and first computations
(iii) Definition of Chain Complexes and their Homology groups
(iv) Singular Homology of spaces and Simplicial Homology of Simplicial Complexes
 (v) Basic Results about Singular Homology: Mayer-Vietoris Theorem, long exact sequence of a pair,
    homotopy invariants, K\"unneth Theorem, Hurewicz Theorem
 (vi) CW complexes and the Homology of the Homology theory; i.e., Axioms determining Homology.
 (vii) Singular Cohomology and its Ring Structure
 (viii) Computations of Singular Homology of various Spaces.

II. Algebraic Topology of Smooth Manifolds
 (i) Sard's Theorem and General Position
 (ii) Morse Functions
 (iii) Poincar\'e Duality
 (iv) Homological intersectiion of cycles
 (v) More Computations

Background Material
Primer on Category theory can be found here.
Primer can on Point Set Topology be found here.
Primer on  limits and colimits can be found here.   
Primer on simplicial complexes can be found here.
Primer on Manifolds can be found here.

Lecture Notes.

Notes for Lectures on Basics  of Homotopy Theory: Sept. 8 - 17 are here.
Notes for Lectures on Basics of Chain Complexes and Homology: Sept. 22 & 24 are here.
Notes for Lectures on Singular Homology: Sept. 29 & Oct.1 are here.
Notes for Lectures on Further Properties of Singular Homology: Oct. 6 , 8, &13 are here.
Notes for Lectures on Singular Cohomology: Oct. 15 & 20 are here.
Notes for Lectures on Pseudomanifolds Representing Homology  Oct. 22-- 27 are here.
Notes for Lectures on the Hurewicz Theorems: Oct. 29, Nov.  & 5 are here.
Notes for Lectures on CW_Complexes:Nov 10 -12 are here
Notes for Lectures on the Homotopy Theory of CW Complexes: Nov. 17 & 19 are here.
Notes for Lectures On the Algebraic Toppology of Smooth Manifolds:: Nov. 24, Dec 3, 8,10 are here.