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!