# Tagged Questions

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

**10**

votes

**1**answer

176 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...

**0**

votes

**0**answers

136 views

### natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$?

is there any natural map from natural map from $H^3(BG,\mathbb{Z})$ to $H^3(\underline E G,\mathbb{Z})$? where $G$ is a discrete group, $BG$ is the classifying space, and $\underline E G$ is the ...

**1**

vote

**0**answers

223 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 ...

**9**

votes

**1**answer

353 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 ...

**3**

votes

**1**answer

180 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 ...

**8**

votes

**2**answers

325 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 ...

**8**

votes

**1**answer

274 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 $E|_{C_+X}\cong C_+X\times G$, ...

**9**

votes

**2**answers

372 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 ...

**10**

votes

**1**answer

343 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 ...

**2**

votes

**0**answers

168 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 ...

**1**

vote

**1**answer

240 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

**1**answer

492 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.
...

**4**

votes

**0**answers

139 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

75 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 ...

**5**

votes

**3**answers

290 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 ...

**3**

votes

**2**answers

354 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

122 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 ...

**7**

votes

**4**answers

587 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 $X$ ...

**4**

votes

**1**answer

156 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

260 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

419 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 ...

**0**

votes

**0**answers

177 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

**8**

votes

**0**answers

358 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 ...

**5**

votes

**2**answers

257 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 $\mathcal{C}_G$ ...

**3**

votes

**2**answers

237 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 ...

**1**

vote

**0**answers

84 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

182 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 ...

**2**

votes

**1**answer

187 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 ...

**2**

votes

**1**answer

213 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 ...

**16**

votes

**2**answers

814 views

### How much of homotopy theory can be done using only finite topological spaces?

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...

**8**

votes

**1**answer

375 views

### Fundamental group of $\mathbb{R}^3-F$ where $F\subseteq \mathbb{R}\times \{0\} \times \{0\}$

Maybe not research level.
Let $Z\cong \mathbb{R}$ be the $z$-axis of $\mathbb{R}^3$. Clearly $\pi_1(\mathbb{R}^3-Z)\cong \mathbb{Z}$. Now if $F\subset Z$ is a closed non-empty subset, then one easily ...

**1**

vote

**1**answer

208 views

### If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if
...

**4**

votes

**2**answers

583 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?

**5**

votes

**1**answer

146 views

### Connection between framed cobordisms and zero sets

Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...

**5**

votes

**0**answers

219 views

### Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where ...

**2**

votes

**0**answers

170 views

### On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...

**14**

votes

**4**answers

441 views

### Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the ...

**11**

votes

**0**answers

348 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**2**

votes

**2**answers

351 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 ...

**9**

votes

**1**answer

263 views

### Representation of finite groups in a compact Lie group

Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then
it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set ...

**3**

votes

**1**answer

164 views

### Carlsson's spectrum BG^-V

In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the ...

**4**

votes

**2**answers

276 views

### $K$-homology of $BG$

Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal.
...

**4**

votes

**1**answer

160 views

### $RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter ...

**11**

votes

**2**answers

314 views

### The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...

**1**

vote

**0**answers

274 views

### Grothendieck's letter to serre

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?

**1**

vote

**0**answers

90 views

### A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof?
Statement. ...

**1**

vote

**1**answer

151 views

### Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a ...

**2**

votes

**3**answers

491 views

### Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?

**1**

vote

**0**answers

173 views

### Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1].
Is there a known tight upper bound in the number of polytopes in ...

**4**

votes

**1**answer

231 views

### Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...