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Is there any example of a "classic" problem in algebraic topology which was solved using "modern" algebraic topology? (i.e. methods using localizations, category-style theories etc.) Just trying to realize why it is all was created and studied.

Disclaimer: I wonder if i had to add some tags referring to non-concrete type of my question. Please edit or help me with it too.

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    Hill-Hopkins-Ravenel resolved the Kervaire invariant one problem, significantly improving our understanding of exotic spheres. – 2017-01-13
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    @MikeMiller: If this is an answer, then you should post it as an answer... – 2017-01-13
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    A good answer to this question should be more than a sentence long. – 2017-01-13
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    @MikeMiller: How about a sentence and a link to a good place where the OP can read more? – 2017-01-13
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    You might enjoy Frank Adams' lovely book *Infinite Loop Spaces*. View it as a report from the battlefield at a time when a lot of the ideas you are interested in where first being created. [May's review](https://projecteuclid.org/download/pdf_1/euclid.bams/1183544574) of the book is entertaining and available on line. – 2017-01-13

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