# Tagged Questions

274 views

### Undergraduate-level intro to homotopy

I'm looking for an undergraduate-level introduction to homotopy theory. I'd prefer a brief (<200pp.) book devoted solely/primarily to this topic. IOW, something in the spirit of the AMS Student ...
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### Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
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### Introductory books as preparation to read Voevodsky homotopy-theory (HoTT) book

I would like to read Voevodsky HoTT book. However, I lack a lot of the basics. I would need a few introductory books first that cover topics like groupoids, fibrations, W -types, Homotopy theory. ...
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### 2 out of 3 axiom and simplicial sets

Let $i\colon\mathcal W\to\mathcal C$ be the inclusion of a subcategory. Unless I'm mistaken, the 2 out of 3 axiom for $\mathcal W$ to be a category of weak equivalences can be expressed as the ...
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### Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
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### The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
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### Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$0 \to bP^{n+1} \to \Theta^n \to Coker(J_n)$$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
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### introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
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### reference request: Postnikov towers for non-simply-connected spaces

I've read that for a space $X$ which is connected but not necessarily simply-connected, we can no longer obtain the $n^{\rm th}$ layer $P_nX$ of the Postnikov tower for $X$ as the pullback of a ...
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### Equivariant homotopy equivalence of based loop group

Consider a compact, connected, simply connected Lie group $G$ and consider $S^1$ as an additive group. Let $\Omega G = \{ \gamma: S^1 \to G: \gamma(0) = e_G\}$ be the corresponding based loop group of ...
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### Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
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### Courses on Homotopy Theory

This autumn I'm considering taking an "advanced" reading course in Algebraic Topology, more specifically homotopy theory. I could extend this reading course over a year and wouldn't mind studying hard ...
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### What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
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### What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
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### Exact Constructions of Homotopy Fiber and Cofiber of Spectra

Given a map of spectra (pick whatever category you want), $f:X\to Y$, what are the exact constructions of the fiber and cofibers of this map? Does this depend in any deep way upon the category or ...
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### Brave New Number Theory

I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question. Is it clear exactly how much (assumedly algebraic) ...
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### Good Reference for Spanier-Whitehead duality?

Does anyone know of a good book that explains Spanier-Whitehead duality (other than Adams)? Thanks Jon