# Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

6,228
questions

**29**

votes

**8**answers

2k views

### Noncommutative rational homotopy type

Ok, this question is much less ambitious than it might sound, but still:
Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-...

**5**

votes

**2**answers

271 views

### Truncated exact sequence of homotopy groups

This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma ...

**35**

votes

**4**answers

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### Chain homotopy: Why du+ud and not du+vd?

When one wants to prove that a morphism $f_*$ between two chain complexes $\left(C_*\right)$ and $\left(D_*\right)$ is zero in homology, one of the standard approaches is to look for a chain homotopy, ...

**110**

votes

**7**answers

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### Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...

**17**

votes

**6**answers

2k views

### Diffeomorphism of 3-manifolds

Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder ...

**28**

votes

**3**answers

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### What manifolds are bounded by RP^odd?

Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...

**152**

votes

**7**answers

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### Proofs of Bott periodicity

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...

**21**

votes

**1**answer

691 views

### The density hex

Gale famously showed that the determinacy of n-player, n-dimensional Hex is equivalent to the Brouwer fixed point theorem in n dimensions.
We can (and Gale does) view this as saying that if you d-...

**2**

votes

**1**answer

445 views

### Are the C(S^n, S^n)'s homeomorphic ?

Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...

**8**

votes

**1**answer

472 views

### Finding cocycles that square to zero

Suppose $x$ is a chosen class in the singular cohomology (integer coefficients) of a space $X$. I'm thinking primarily of classes of odd degree on a simply connected space. What are necessary ...

**15**

votes

**1**answer

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### Graded commutativity of cup in Hochschild cohomology

I am trying to get used to Hochschild cohomology of algebras by proving its properties. I am currently trying to show that the cup product is graded-commutative (because I heard this somewhere); ...

**8**

votes

**5**answers

4k views

### (infinity,1)-categories directly from model categories

Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" ...

**18**

votes

**6**answers

2k views

### Simplicial homotopy book suggestion for HTT computations

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I'...

**56**

votes

**10**answers

15k views

### Nice proof of the Jordan curve theorem?

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive ...

**4**

votes

**2**answers

561 views

### HNN extensions which are free products

which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...

**3**

votes

**3**answers

673 views

### Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...

**15**

votes

**2**answers

695 views

### Spin structures on 7-dimensional spherical space forms

Background
Let $M$ be a spin manifold and let $\Gamma$ be a finite group acting freely and isometrically on $M$ in such a way that $M/\Gamma$ is a smooth riemannian manifold. The quotient will be ...

**2**

votes

**3**answers

690 views

### Two solid N_3 glued by its boundary

Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d ...

**25**

votes

**6**answers

3k views

### Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...

**12**

votes

**1**answer

714 views

### Smooth structures on PL 4-manifolds

Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL ...

**5**

votes

**2**answers

347 views

### Connectivity after Geometric Realization?

Suppose that I have a map of simplicial spaces,
$ f: X_* \to Y_*$,
and that I know that the map on zero spaces $f_0: X_0 \to Y_0$ is n-connected. Can I conclude anything about the connectivity of ...

**2**

votes

**2**answers

867 views

### Periodic mapping classes of the genus two orientable surface

Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...

**21**

votes

**9**answers

3k views

### What methods exist to prove that a finitely presented group is finite?

Suppose I have a finitely presented group (or a family of finitely presented groups with some integer parameters), and I'd like to know if the group is finite. What methods exist to find this out? I ...

**34**

votes

**21**answers

5k views

### Generalizations of Planar Graphs

This is a follow up to Harrison's question: why planar graphs are so exceptional. I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; ...

**16**

votes

**2**answers

2k views

### Topologically contractible algebraic varieties

From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...

**6**

votes

**2**answers

535 views

### Naive Z/2-spectrum structure on E smash E?

Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...

**32**

votes

**2**answers

8k views

### Explanation for the Thom-Pontryagin construction (and its generalisations)

In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...

**27**

votes

**4**answers

4k views

### The Jouanolou trick

In Une suite exacte de Mayer-Vietoris en K-théorie algébrique (1972) Jouanolou proves that for any quasi-projective variety $X$ there is an affine variety $Y$ which maps surjectively to $X$ with ...

**5**

votes

**2**answers

756 views

### What is $TC(\Sigma^\infty \Omega X)$?

I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and ...

**10**

votes

**1**answer

829 views

### Cyclic spaces and S^1-equivariant homotopy theory

I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...

**4**

votes

**3**answers

895 views

### Computation of Joins of Simplicial Sets

It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...

**1**

vote

**1**answer

231 views

### N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms

It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov ...

**37**

votes

**3**answers

3k views

### Topological Langlands?

In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...

**7**

votes

**3**answers

815 views

### Joins of simplicial sets

Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at ...

**45**

votes

**6**answers

5k views

### Why is the standard definition of cocycle the one that _always_ comes up??

This question might not have a good answer. It was something that occurred to me yesterday when I found myself in a pub, needing to do an explicit calculation with 2-cocycles but with no references ...

**27**

votes

**4**answers

5k views

### What (if anything) happened to Intersection Homology?

In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...

**22**

votes

**1**answer

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### Characteristic classes of sphere bundles over spheres in terms of clutching functions

I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...

**6**

votes

**4**answers

896 views

### What is known about the intersection pairing on H^{mid}?

When we restrict to the torsion-free part of the cohomology of a manifold, the intersection pairing is nondegenerate. In dimension 2n, this gives a bilinear form on the free part of Hn (symmetric if ...

**10**

votes

**2**answers

2k views

### The De Rham Cohomology of $\mathbb{R}^n - \mathbb{S}^k$

I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots.
Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show ...

**0**

votes

**1**answer

262 views

### Homology of symmetric groups

Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$?

**50**

votes

**10**answers

7k views

### de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...

**8**

votes

**3**answers

1k views

### How are multiplicative sequences related to formal power series and genera of manifolds?

Let $B$ be the graded ring $\bigoplus_i B^i$ (with $B^k B^l \subset B^{k+l}$), and $B_f$ the multiplicative group of all formal sums $1 + b_1 + b_2 + \cdots$ where $b_i \in B^i$ for all $i$.
The idea ...

**3**

votes

**1**answer

318 views

### Is the coproduct of fibrant spectra fibrant again?

Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...

**9**

votes

**3**answers

913 views

### Integration in equivariant K-theory

Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the ...

**40**

votes

**6**answers

5k views

### Why does one think to Steenrod squares and powers?

I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...

**12**

votes

**2**answers

1k views

### Combinatorics of the Stasheff polytopes

First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...

**1**

vote

**1**answer

802 views

### Simplicial set notation and vocabulary question.

Notation question:
What does $(\Delta^1)^{ \{1, \ldots,n-1 \}}$ denote? UPDATE: I (David Speyer) tried to fix the LaTeX. Please see if I got it right.
Vocabulary question:
Suppose $z:\Delta^{n+1} \...

**46**

votes

**10**answers

11k views

### Definition of “simplicial complex”

When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".
However, ...

**66**

votes

**11**answers

10k views

### What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...

**5**

votes

**1**answer

509 views

### Spanier-Whitehead dual and Hopf fibration

Consider a map of spheres $f:S^n\to S^m$ covered by a map of trivial $\mathbb R^k$-bundles.
In other words, we take the trivial rank $k$ vector bundle over $S^m$ and pull it to $S^n$ via $f$. Consider ...