75 votes
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There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

I once asked André Weil the same question. When I was college, taking a course that discussed quadratic forms, Weil gave a guest lecture to the students about that topic. After the talk, I raised my ...
KConrad's user avatar
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72 votes
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In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

Here is an example of topological space $X$, embeddable as compact subspace of $\mathbf{R}^3$, that is not simply connected, but in which every simple loop is homotopic to a constant loop. Namely, ...
YCor's user avatar
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59 votes

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

The MathSciNet review (by Julie Bergner) of Simpson's book: Homotopy theory of higher categories, New Mathematical Monographs, 19. Cambridge University Press, Cambridge, 2012, has a note about the ...
Edward Dunne's user avatar
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52 votes
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What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan ...
Yonatan Harpaz's user avatar
52 votes
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Spheres with the same homotopy groups

It is a result from Serre's thesis that for $n\geq 3$ and a prime $p$, the first $p$-torsion in $\pi_*S^n$ occurs precisely for $* = n+2p-3$. This shows that $(m,n) = (2,3)$ is the only pair of (edit: ...
Lennart Meier's user avatar
50 votes

What is modern algebraic topology(homotopy theory) about?

While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still ...
44 votes

Timeline of "foundational" advances in homotopy theory?

Such a timeline is necessarily highly subjective. With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory. The list below concentrates ...
42 votes

Why is the definition of the higher homotopy groups the "right one"?

I think that obstruction theory is one of the most important reasons to study homotopy groups. If you are interested in studying the possible homotopy classes of maps $X \to Y$ of spaces where $X$ has ...
41 votes

Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$

The fundamental group of the one-point compactification of $\mathbf R^3 \setminus \{p_1,\ldots,p_n\}$ is a free group on $n$ generators, for any $n \geq 0$.
Dan Petersen's user avatar
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40 votes
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Morava K-theories for dummies?

This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H_*(X)$, the ordinary homology of $X$. Later ...
Neil Strickland's user avatar
40 votes

In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

Every finite simplicial complex is weakly homotopy equivalent to a finite space. Therefore there are finite spaces with nontrivial loops; and these are obviously not embedded.
Jeff Strom's user avatar
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40 votes
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Mark Hovey's open problems in the theory of model categories

I am a former student of Mark Hovey's, and during grad school, I wrote a document giving an update on the status of the 13 problems (as of 2012 or 2013, I guess). I just briefly went through it a ...
David White's user avatar
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39 votes

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer. First I entirely agree with Yonatan that the main problem ...
Simon Henry's user avatar
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38 votes

What is the 31st homotopy group of the 2-sphere?

My apologies for updating this very old question. As already mentioned, the 31st homotopy group of $S^2$ is the same as the 31st homotopy group of $S^3$. Serre's mod-C theory shows that this is a ...
Tyler Lawson's user avatar
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38 votes
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The homotopy category is not complete nor cocomplete

I'll work in the based category, and consider $S^1$ as $\{z\in\mathbb{C}:|z|=1\}$. Consider the maps $$\text{point}\xleftarrow{}S^1\xrightarrow{f}S^1, $$ where $f(z)=z^2$. Suppose that there is a ...
Neil Strickland's user avatar
38 votes
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What is a triangle?

To answer your first precise questions: Yes, every distinguished triangle in $D(A)$ comes from a short exact sequence. For every distinguished triangle $X \to Y \to \mathrm{Cone}(f) \stackrel{+1}\to $...
Dan Petersen's user avatar
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38 votes

Why do we need model categories?

Model categories capture the idea that in many cases you resolve an object by an equivalent object that is better behaved. The standard example is replacing a chain complex by a chain complex of ...
Lennart Meier's user avatar
36 votes
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Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme

The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) ...
Simon Henry's user avatar
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36 votes
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Higher Topos Theory- what's the moral?

It seems there are really two questions here: Why higher category theory? What questions can you pose without the language of higher category theory which are best answered using higher category ...
Tim Campion's user avatar
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35 votes

What is the cotangent complex good for?

Here is an example mentioned in passing by user ali's answer, but I think it is cute (and powerful) enough to be worth fleshing out the details. Lifting from characteristic $p$ to characteristic zero ...
Piotr Achinger's user avatar
34 votes

Analogue to covering space for higher homotopy groups?

My apologies for coming back to this old question, but I want to address a point that I think is not really addressed so far. Namely, for $n=1$, the universal cover of a (reasonable) topological space ...
Peter Scholze's user avatar
33 votes
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What is the relationship between connective and nonconnective derived algebraic geometry?

Here is an example of a nonconnective $E_\infty$ ring spectrum which, I think, illustrates a key problem. (A more extensive discussion of this phenomenon occurs in Lurie's DAG VIII and in a paper by ...
Tyler Lawson's user avatar
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33 votes
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What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

Lurie characterizes the symmetric monoidal structure on $\mathsf{Sp}$ by a universal property (HA.4.8.2.19): it is uniquely determined up to a contractible space of choices by the property that $S^0$ ...
Dylan Wilson's user avatar
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32 votes

References and resources for (learning) chromatic homotopy theory and related areas

Preliminaries (i.e. Advanced Algebraic Topology) General References Advanced Algebraic Topology, Alexander Kupers; More Concise Algebraic Topology, J. Peter May; Introduction to Homotopy Theory, Paul ...
31 votes
Accepted

Endomorphism ring spectrum of the Eilenberg-MacLane spectrum

No, $A$ is not an $H\Bbb Z$-algebra. Suppose $R$ is an $H\Bbb Z$-algebra. Then the category of left $R$-modules is $H\Bbb Z$-linear: for any $R$-modules $M$ and $N$, the function spectrum $F_R(M,N)$ ...
Tyler Lawson's user avatar
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31 votes
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An equivariant map from sphere to a Lie group of lower dimension which is not null homotopic?

The Blakers-Massey element in $\pi_6(S^3)\cong\mathbb{Z}_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper https://arxiv.org/abs/math/0501091, published as ...
Mark Grant's user avatar
31 votes
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Why is Voevodsky's motivic homotopy theory 'the right' approach?

(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understand ...
Denis Nardin's user avatar
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31 votes
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If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic

The result you are hoping for is in fact false. In section 9 of Microbundles: Part I, Milnor constructs an open set $U \subset \mathbb{R}^m$. With its standard smooth structure, the (stable) tangent ...
Michael Albanese's user avatar
31 votes
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Why are quasi-categories better than simplicial categories?

As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it ...
Tim Campion's user avatar
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