Algebraic Topology Book for the Analyst I'm looking for a graduate level book on topology that takes most of its motivation from analysis and applied mathematics. I'm currently in an algebraic topology course but other than basic definitions and intuition I have learned absolutely nothing about algebraic topology!
Now I need to go and relearn most of the topic but it's quite challenging because I find most books on topology unmotivated and uninteresting. Most examples in my class are showing that one sphere is not a different sphere or otherwise use a collection of classic topological objects which I take very little interest in.
I am not trying to diss algebraic topology in any way, but what are some books on topology that stress spaces that are of greater interest to problems in analysis and probability theory?
Ghrist's book "Elementary Applied Topology" looks good, but too cursory for what I'm after. And references on topological data analysis use persistent homology and other topics that are currently above my head.
Some books that I'm aware of but have not read that seem like they may be good are: Lee "Introduction to Topological Manifolds", Dold "Lectures in Algebraic Topology", Rotman "An Introduction to Algebraic Topology", Edelsbrunner "Computational Topology", and Kaczynski "Computational Homology"
If one of the above texts stands out as a good candidate for what I'm interested in, please let me know (It's impossible to read all of them before deciding).
Books that I have read parts of and dislike include: Bredon, Massey, Hatcher, and May.
 A: Since you say in the post that you only learned nothing but basic definitions, intuitions and some basic examples, I guess you also need some problem book?
For pure algebraic topology, e.g. homology, cohomology, etc, you could directly follow the exercises in

Hatcher's "Algebraic Topology".

There are several solution manuals free online addressing the exercises in homology and cohomology, and some from homotopy theory, fundamental group, etc.
But note that Hatcher talked about homology firstly from simplicial homology, so many exercises are about how to construct a simplex in a given space, which is really geometric... 
This is why I don't like Hatcher's book. I like starting from cellular homology which is more general, and reduced down to simplicial homology. 

If you want some algebraic topology exercise that addresses in analysis and probability theory (so I guess you also mean some PDE?), then you will need a book or a problem book containing exercises and solutions on differentiable manifolds. For this, I recommend 

"Analysis and Algebra on Differentiable Manifolds --- A Workbook for students and teachers", by Pedro M. Gedea, Jaime Munoz Masque and Ihor V. Mykytyuk.

This books contains everything you need from first year graduate differentiable manifolds to Riemannian geometry.
This book is a problem book, with detailed solutions. It also summarizes some important definitions and theorems at the beginning of each sections, so if you have been familiar with some notions, this book is really good for exercises. 
The exercises are half analysis and half algebraic, so I believe you will like it.
I think google-book allows you to read many pages of it, so you could take a look and have a taste of it.
Hope this help :)
