Prerequisites for Gunnar Carlsson's Topology and Data I am planning on doing a project on topological data analysis in the near future and intend to use Gunnar Carlsson's paper "Topology and Data" as my introduction to the field. I am familiar with point-set topology and some differential geometry but know little about algebraic topology which is the mathematical foundation of this branch. My question therefore is, what topics in algebraic topology should I study beforehand? 
I have looked around on the internet but can't find a list of topics I should be acquainted with. Scanning the paper itself I've noticed some relevant definitions and theorems are given but I presume the paper is not self-contained. Furthermore I noticed a theorem (2.4, pg. 9) involving Riemannian manifolds. Will I need any Riemannian geometry?
A related question, to what type of data set are the methods described in this paper particularly suited? Is there an easy to understand example? I can't be more specific about this last question because obviously I have not yet read  the paper. I am looking for a general answer.
Thanks in advance!
 A: As I have said in the comments, for sure it would help to know the basics of Homology (I have seen that the author also mention homotopy and the $\pi_1$ group but I assume that you already have seen these things). 
The problem is that those texts that I mentioned above (Hatcher and Matveev) are a good introduction but from another point of view maybe are too much. 
I mean that it is good to know simplicial, singular and cellular homology but probably to understand the paper you don't need all of them. You should look into something more oriented to topological data analysis.
So, if you want to build solid basis for future studies in Topology, then it would be a good idea to read for example the first of the two  chapters of Matveev, maybe skipping the proofs and focusing on the definitions*.
I think that Matveev's book it is a good trade off between conciseness and  doing all the steps. 
The paper seems "almost" self-contained, it defines what is an abstract simplicial complex, Cech complex and persistent homology.
Maybe it would be a good thing to have a more detailed text on these topics.
I have found this  interesing thesis of Brian Brost
http://www.math.ku.dk/~moller/students/brian_brost.pdf
The first two chapters seems to expand nicely the over mentioned topics.
I hope it helps.
*Sections from 1.1 to 1.6  are fundamentals (i.e. IMHO good to build some knowledge about simplicial homology) as sections from 1.9 to 1.14., the rest is probably even more far from your actual aim. 
