Elementary Applied Topology by Robert Ghrist.
From the introduction:
What topology can do
Topology was built to distinguish qualitative features of spaces and mappings. It is good for, inter alia:
- Characterization: Topological properties encapsulate qualitative signatures. For example, the genus of a surface, or the number of connected components of an object, give global characteristics important to classification.
- Continuation: Topological features are robust. The number of components or holes is not something that should change with a small error in measurement. This is vital to applications in scientific disciplines, where data is never noisy.
- Integration: Topology is the premiere tool for converting local data into global properties. As such, it is rife with principles and tools (Mayer-Vietoris, Excision, spectral sequences, sheaves) for integrating from local to global.
- Obstruction: Topology often provides tools for answering feasibility of certain problems, even when the answers to the problems themselves are hard to compute. These characteristics, classes, degrees, indices, or obstructions take the form of algebraic-topological entities.
What topology cannot do
Topology is fickle. There is no resource to tweaking epsilons should desiderata fail to be found. If the reader is a scientist or applied mathematician hoping that topological tools are a quick fix, take this text with caution. The reward of reading this book with care may be limited to the realization of new questions as opposed to new answers. It is not uncommon that a new mathematical tool contributes to applications not by answering a pressing question-of-the-day but by revealing a different (and perhaps more significant) underlying principle.
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The text will require more than casual interest but what a tool to add to your toolbox!
I first saw this in a tweet from Topology Fact.