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

**6**

votes

**2**answers

417 views

### Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves

Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper ...

**44**

votes

**11**answers

5k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...

**11**

votes

**2**answers

2k views

### Euler characteristic of a manifold and self-intersection

This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...

**49**

votes

**6**answers

5k views

### third stable homotopy group of spheres via geometry?

It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...

**8**

votes

**1**answer

2k views

### Kuenneth-formula for group cohomology with nontrivial action on the coefficient

For a trivial action on the coefficient, we have the following Kuenneth formula
for group cohomology:
$$
H^n(G_1 \times G_2; M) \cong
[\oplus_{i= 0}^n H^i(G_1;M) \otimes_M H^{n-i}(G_2;M)]
\oplus ...

**18**

votes

**3**answers

2k views

### finite generated group realized as fundamental group of manifolds

This is discussed in the standard textbooks on algebraic topology.
Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$
where $g_i$ are generators and $r_j$ are ...

**12**

votes

**6**answers

2k views

### Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here.
Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$.
Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...

**25**

votes

**2**answers

5k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see this related post and also the following post.
Added : According to their ...

**86**

votes

**6**answers

5k views

### Counterexamples in algebraic topology?

In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...

**52**

votes

**9**answers

4k views

### What are the uses of the homotopy groups of spheres?

Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:
Have the homotopy groups of spheres ever been applied to ...

**36**

votes

**6**answers

4k views

### Cubical vs. simplicial singular homology

Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old ...

**47**

votes

**8**answers

5k views

### What is a continuous path?

I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...

**18**

votes

**5**answers

6k views

### How do you show that $S^{\infty}$ is contractible?

Here I mean the version with all but finitely many components zero.

**32**

votes

**1**answer

1k views

### Brown representability for non-connected spaces

In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this:
If $F$ ...

**16**

votes

**4**answers

2k views

### Fundamental groups of topological groups.

Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is:
If $G$ is a ...

**18**

votes

**1**answer

531 views

### What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$?

What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$? The group structure induced by the group structure on the codomain. This question is a followup to Eric's answer ...

**16**

votes

**7**answers

2k views

### Whitehead for maps

I made the following claim over at the Secret Blogging Seminar, and now I'm not sure it's true:
Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same ...

**12**

votes

**1**answer

899 views

### finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...

**5**

votes

**1**answer

279 views

### Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...

**66**

votes

**10**answers

12k views

### Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...

**82**

votes

**18**answers

8k views

### Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...

**59**

votes

**7**answers

7k views

### 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, ...

**73**

votes

**6**answers

4k views

### What properties make $[0,1]$ a good candidate for defining fundamental groups?

The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...

**18**

votes

**5**answers

4k views

### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

**36**

votes

**6**answers

4k views

### Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...

**43**

votes

**4**answers

3k views

### Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...

**34**

votes

**3**answers

4k views

### Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...

**28**

votes

**8**answers

6k views

### An “advanced beginner's” book on algebraic topology?

It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read ...

**35**

votes

**2**answers

2k views

### Are spectra really the same as cohomology theories?

Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. ...

**32**

votes

**7**answers

3k views

### What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello,
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :
For example :
1) Are all ...

**23**

votes

**4**answers

2k views

### What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of ...

**32**

votes

**20**answers

4k 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; ...

**9**

votes

**4**answers

1k views

### Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...

**27**

votes

**5**answers

3k views

### Fundamental groups of noncompact surfaces

I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...

**24**

votes

**2**answers

2k views

### generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...

**19**

votes

**2**answers

1k views

### Converse to Stokes' Theorem

Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : ...

**10**

votes

**3**answers

1k views

### When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...

**29**

votes

**4**answers

2k views

### Integral cohomology (stable) operations

There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...

**19**

votes

**3**answers

2k views

### “Dirty” proof that Eilenberg-MacLane spaces represent cohomology?

The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...

**16**

votes

**5**answers

2k views

### Maps inducing zero on homotopy groups but are not null-homotopic

Today my fellow grad student asked me a question, given a map f from X to Y, assume $f_*(\pi_i(X))=0$ in Y, when is f null-homotopic?
I search the literature a little bit, D.W.Kahn
...

**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 ...

**20**

votes

**0**answers

460 views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...

**15**

votes

**2**answers

1k views

### (co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...

**24**

votes

**2**answers

1k views

### What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...

**15**

votes

**3**answers

1k views

### Is there Domain Invariance for Alexandrov spaces?

A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for ...

**14**

votes

**3**answers

1k views

### When are (finite) simplicial complexes (smooth) manifolds?

Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?

**19**

votes

**2**answers

943 views

### Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book ...

**15**

votes

**4**answers

953 views

### Details for the action of the braid group B_3 on modular forms

I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...

**12**

votes

**7**answers

2k views

### Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...

**26**

votes

**1**answer

529 views

### Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...