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

**14**

votes

**0**answers

361 views

### Are there “chain complexes” and “homology groups” taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A ...

**5**

votes

**0**answers

57 views

### Nonunital $E_\infty$-rings

An elementary fact of algebra is that the category of nonunital commutative rings is equivalent to that of $\mathbb{Z}$-augmented unital commutative rings, the equivalence being given by forming ...

**0**

votes

**0**answers

69 views

### Higher Homotopy Groups

Theorem 5.1 of this paper
describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...

**3**

votes

**1**answer

58 views

### Fixed point relation $\ Fix\ $ for pairs of manifolds

First the classical definition: a topological space $X$ has the fixed point property (fpp) $\ \Leftarrow:\Rightarrow\ $ for every continuous $\ f : X\rightarrow X\ $ there exists $\ p\in X\ $ such ...

**3**

votes

**0**answers

65 views

### Inverse cohomological isomorphisms

Let $\ M'\ M''\ $ be simply-connected Hausdorff compact manifolds (possibly with boundary for another variant of the question). Let $\ f:M'\rightarrow M''\ $ be a continuous function which induces an ...

**4**

votes

**3**answers

253 views

### In a fibration, can a deformation retraction of the base be lifted to the total space?

Given a fibration $p:E \rightarrow B$ and if $A$ is a deformation retract of $B$. Is it true that $p^{-1}(A)$ is a deformation retract of $E$?. If this is not true, can some conditions be imposed on ...

**4**

votes

**0**answers

174 views

### What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem.
Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...

**3**

votes

**0**answers

66 views

### Clutching functions and Classifying maps

Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, ...

**7**

votes

**1**answer

213 views

### How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map
$$
\mu: G\times ...

**4**

votes

**1**answer

167 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**8**

votes

**1**answer

225 views

### Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...

**2**

votes

**2**answers

289 views

### RO(G) grading of Mackey functors

If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational ...

**8**

votes

**1**answer

300 views

### Classifying TQFTs with 1d vector spaces

To what extent have people classified $n$-dimensional TQFTs that assign a 1-dimensional vector space to every compact oriented $(n-1)$-manifold?
I have some vague reasons to suspect that the ...

**5**

votes

**2**answers

589 views

### How nilpotent is the ring of stable homotopy groups of spheres?

Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?

**1**

vote

**1**answer

201 views

### pontryagin dual and maps between spectra

Given two spectra $A$ and $B$, the set $[A,B]$ of homotopy classes of maps from $A$ to $B$ forms an abelian group. Can the dual abelian group $\text{Hom}([A,B],\mathbb{Q}/\mathbb{Z})$ be expressed as ...

**7**

votes

**4**answers

548 views

### Constructing a space with prescribed cohomology ring

The most general way I can formulate my question is the following:
Question 1: Given a Gorenstein quotient ring $S$ of a polynomial ring over a field $K$, can one construct a (topological) space $X$ ...

**7**

votes

**1**answer

387 views

### Learning roadmap to TQFT from a mathematics perspective

I had asked a question on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posting it below.
...

**2**

votes

**0**answers

144 views

### Bott's Formula for Grassmannians

Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space ...

**8**

votes

**5**answers

2k views

### Are there two non-homotopy equivalent spaces with equal homotopy groups?

Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...

**7**

votes

**1**answer

195 views

### Cosimplicial commutative rings in stable homotopical algebra

When doing "derived commutative algebra" over a discrete commutative ring $R$ that doesn't contain $\mathbb{Q}$, it's fairly well known that you generally have two flavors of "totally commutative ...

**15**

votes

**1**answer

702 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**5**

votes

**1**answer

326 views

### A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here:
http://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property
Assume that $0<k<n-1$, ...

**10**

votes

**1**answer

503 views

### Reference request for TQFT, functoriality

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, ...

**2**

votes

**0**answers

96 views

### On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...

**1**

vote

**0**answers

71 views

### Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...

**3**

votes

**2**answers

325 views

### Zigzags and contractibility of categories

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:
If there is ...

**1**

vote

**1**answer

106 views

### homotopy type of the cone of a loop space

I read somewhere that for, a path connected CW complex $X$, there is a homotopy equivalence of pairs between $(P_1X,\Omega X)$ and $(C\Omega X,\Omega X)$ where $P_1X$ denotes the set oh paths ...

**2**

votes

**1**answer

156 views

### simple explanation of simplicial volume=4g-4 when genus $\ge 1$

In Gromov's famous book, it says "simplical volume of every oriented surface of genus $ \ge 1$ satisfies ${\left\| {\left[ S \right]} \right\|_\Delta } = 4g - 4 = - 2\chi \left( S \right) = - ...

**8**

votes

**0**answers

447 views

### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book Lectures on tensor categories and modular functors by Bakalov ...

**14**

votes

**2**answers

398 views

### Is there an effective way to calculate K-theory using Morse functions?

Let $M$ be a compact manifold and let $f$ be a Morse function with exactly one critical point at each critical level. Then one can recover a CW-complex with the homotopy type of $M$ from just the ...

**4**

votes

**1**answer

129 views

### AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article
Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386)
(theorems 6.11 and ...

**5**

votes

**2**answers

241 views

### Affine structures

I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy ...

**3**

votes

**1**answer

266 views

### Delooping in homotopy type theory

In algebraic topology, it is a theorem of Stasheff that every A-$\infty$ space has the homotopy type of a loop space.
Question: Is this true in homotopy type theory?
Let me be a little more ...

**6**

votes

**1**answer

600 views

### Cartan 3-form on a Lie group G

Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW term is nothing more than the integral of the pullback of the Cartan $3$-form via ...

**10**

votes

**2**answers

1k 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 : Most interesting mathematics mistake?
Added : According to their method, ...

**8**

votes

**0**answers

314 views

### Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...

**0**

votes

**0**answers

163 views

### K-theory of $\mathbb{RP}^\infty$

can anyone give some reference of K-theory and K-homology of $\mathbb{RP}^\infty$, both $K_0$ and $K_1$.
PS: also posted in stackexchange

**5**

votes

**2**answers

229 views

### Is the category of $G$-spaces a model category?

Let $G$ be a compact Lie group and $\mathcal{C}_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}_G$ ...

**4**

votes

**2**answers

514 views

### Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?

**3**

votes

**2**answers

207 views

### Pseudo-manifolds and homology

Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group?
I was looking for a more geometrical definition of homology and ...

**5**

votes

**2**answers

1k views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**1**

vote

**0**answers

79 views

### Thom-Pontryagin construction for pairs

Is there a generalization of the Thom-Pontryagin construction in the following sense? Let $M$ be a smooth manifold, $\partial M=A\cup B$ where $A$ and $B$ are $m-1$-manifolds with a common boundary ...

**3**

votes

**1**answer

159 views

### Kan condition in simplicial homotopy theory

I know Kan condition (see http://en.wikipedia.org/wiki/Kan_fibration) is something like homotopy extension condition, and I know this condition ensures homotopy defined by the naive idea to be an ...

**27**

votes

**6**answers

2k 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 ...

**7**

votes

**3**answers

429 views

### When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...

**2**

votes

**1**answer

177 views

### Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it weak equivalence?

Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon ...

**16**

votes

**2**answers

2k views

### Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...

**10**

votes

**2**answers

716 views

### Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...

**9**

votes

**1**answer

363 views

### Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...

**2**

votes

**1**answer

196 views

### Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $h\circ i:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...