Posted by Jason Polak on 25. March 2015 · Write a comment · Categories: algebraic-topology, books

Doug Ravenel has made his book Nilpotence and periodicity in stable homotopy theory available for free download along with a list of errata, also available at the same page as the book.

Here is the official description from Princeton University Press:

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic.

Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

At the official page of the book, you can also buy a paperback copy.

# A is Homotopy Equivalent to A^op via Functors

Let $\mathcal{A}$ be a small category and $\mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $\mathbf{B}\mathcal{A}$ and $\mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen's paper Higher Algebraic K-Theory: I", so grab the notes or read the summary below:

Given any functor $f:\mathcal{A}\to\mathcal{B}$, and an object $B\in\mathcal{B}$, we can consider the category $f^{-1}(B)$ consisting of objects $A\in \mathcal{A}$ such that $f(A) = B$. The morphisms of $f^{-1}(B)$ are defined to be all the morphisms that map to $1_B$ under $f$. Let us apply this to the following situation:

Given any small (or skeletally small) category $\mathcal{A}$, we can construct another category $S(\mathcal{A})$ as follows: the objects of $S(\mathcal{A})$ are the arrows $X\to Y$ of $\mathcal{A}$, and a morphism $(X\to Y)\to (X'\to Y')$ is a pair of morphisms $X'\to X$ and $Y\to Y'$ in $\mathcal{A}$ making the obvious square commute. Now, we can consider the functor $s:S(\mathcal{A})\to \mathcal{A}$ sending the object $X\to Y\in S(\mathcal{A})$ to the object $X\in\mathcal{A}$.
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