## Introduction to algebraic topology

### MA 572- Spring 2021, Prof. Manuel Rivera, Purdue University

This course will be an introduction to algebraic topology. The main topics to be covered will be:

1) singular homology

2) computational methods based on cellular homology of CW complexes

3) the Eilenberg-Zilber and Alexander-Whitney maps, Universal Coefficient Theorem, and Kunneth Theorem

4) cohomology groups and cup product

5) Poincaré duality of manifolds

Along the way will introduce some basic category theory, simplicial techniques, and develop the necessary homological algebra. These tools were born in the context of algebraic topology but are widely used throughout modern mathematics.

**Instructor:** Prof. Manuel Rivera (manuelr at purdue dot edu)

**Textbooks:** The lectures will not strictly follow any textbook. They will be roughly based on these notes written by Haynes Miller:

http://math.mit.edu/~hrm/papers/lectures-905-906.pdf

However, it is highly recommended to complement the lectures by reading the following textbooks:

1) Algebraic Topology, by A. Hatcher

2) Elements of Algebraic Topology, by J. Munkres,

3) Topology and Geometry, G. Bredon

4) Peter May, A Concise Course in Algebraic Topology,

**Pre-requisites:** We will assume familiarity with basic point set topology, covering spaces, and the fundamental group, as discussed in MA571. We will also assume some basic results in algebra such as the classification theorem of finitely generated abelian groups.

**Grading:**** **Homework 75 %, one exam 25%

**Grader/TA****:** The TA for this course is Daniel Tolosa: dtolosav at purdue dot edu

**Office Hours:** Tuesdays after class at 10:30am or by appointment

**Special accommodations:** Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.

In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php

**Homework:**

**Homework:**

Due Firday January 29: 1) Exercises 1.8, 2.2., 2.3, 3.7,3.8 from HM's notes 2) Prove that the first singular homology group of a path-connected space is isomorphic to the abelianization of its fundamental group at any point.

Due Friday February 5: Exercises 4.10, 5.15, and 6.3 from HM's notes.

Due Friday Februrary 26: Exercises 7.3, 8.8, 9.8, 9.10 from HM's notes as well as the following:

Problem 1: Let A —> B —> C —> D —> E be an exact sequence of abelian groups. Prove that C=0 if and only if the map A—> B is surjective and D—>E is injective. Deduce that for all pairs of spaces (A,X), the inclusion A —> X induces isomorphisms on all homology groups if and only if H_n(X,A)=0 for all n. Problem 2: Homology does not determine the fundamental group: Give an example of two spaces that have isomorphic homology groups but non-isomorphic fundamental groups.

For the following three computations you may use any of the theorems proved in class (Mayer-Vietoris, long exact sequence of a pair, homotopy invariance homology, excision,…).

1. Compute the homology groups of $M_g$ the orientable compact surface of genus $g$.

2. Suppose $X$ and $Y$ are two pointed spaces in which each of the base points is a deformation retract of an open set.

Consider the wedge $X \vee Y$ obtained by identifying the base points of $X$ and $Y$. Compute the homology groups of $X \vee Y$ in terms of the homology groups of $X$ and the homology groups of $Y$.

3. Compute the homology groups of the projective plane $\mathbb{R}P^2$ and the Klein bottle $K$.

4. Let $X$ be the space obtained from removing $n$ distinct points from a torus. Compute the homology groups of $X$.

5. Let $X$ be a topological space. Define the suspension of $X$, denoted by $SX$, as the quotient of $X \times [0,1]$ by the relation generated by $(x,1) ~ (x’,1)$ and $(x,0) ~ (x’,0)$. Compute the homology of $SX$ in terms of the homology of $X$.

6. Also do exercise 13.6 from the notes.

Due April 8:

1) Exercise 17.2 HM’s notes- this will not be graded, a solution may be find in Hatcher’s book.

2) For X a finite CW complex and F a field, show that the Euler characteristic of X is given by the alternating sum of the dimensions of the vector spaces H_n(X;F).

3) For finite CW complexes X and Y show that the Euler characteristic of X \times Y is the product of the Euler characteristic of X with the Euler characteristic of Y.

4) Exercise 18.7 HM’s notes

5) Exercise 20.12 HM’s notes

6) a) Consider the space X obtained by identifying pairs of antipodal points in the equator S^1 of S^2. Namely, X= S^2/ ~ where x ~ -x for x in the equatorial S^1. Compute the homology of X. b) Let Y be a similar space constructed from S^3. Namely Y = S^3/ ~ where y ~ -y for y in the equatorial S^2. Compute the homology of Y.

7) Use cellular homology to compute the homology of the quotient space RP^n / RP^k for k < n. Hint: Use the standard CW complex of RP^n which has RP^k as its k-skeleton.