# Tagged Questions

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

**8**

votes

**0**answers

166 views

### What is known about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...

**8**

votes

**1**answer

292 views

### Is the natural isomorphism $|FX_\bullet| \cong F|X_\bullet|$ lax symmetric monoidal?

Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...

**7**

votes

**0**answers

93 views

### Homology of inverse limits over inverse systems more complicated than towers

Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...

**1**

vote

**1**answer

246 views

### $G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? [closed]

As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true ...

**6**

votes

**0**answers

56 views

### Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the cofiber doesn't increase?

Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One ...

**5**

votes

**2**answers

548 views

### Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...

**2**

votes

**0**answers

86 views

### Does attach-one-cell have a stable homotopy transfer?

Specifically, I am thinking of attaching one ordinary cell to an ordinary space in your favourite convenient category of spaces; so, given a cofiber sequence
$$ \mathbb{S}^k \to_c X \to_p X', $$
on ...

**10**

votes

**0**answers

124 views

### Existence of flat connections via characteristic classes, for nice groups

I have two questions about what I write below (which honestly seems pretty elementary).
Is it true (more or less)?
Is there a clean reference that I can cite.
Let $G$ be a compact Lie group, $M$ a ...

**4**

votes

**0**answers

65 views

### Criterion for a equalizer to be a homotopy equalizer in spaces

Let $f,g\colon X\rightarrow Y$ be maps between spaces.
I am looking for criteria for the equalizer of $f$ and $g$ to be a homotopy equalizer and I am happy to get answers for whatever model category ...

**8**

votes

**1**answer

145 views

### Cohomology operations on unoriented cobordism

In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...

**6**

votes

**1**answer

207 views

### Can the Hochschild cochain complex be given the structure of a “homotopy BV algebra”?

In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's):
Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative ...

**2**

votes

**0**answers

68 views

### Can relative homotopy groups be represented as relative homology groups of some Moore complex?

Daniel Kan defined a combinatorial version of the homotopy group $\pi_n(X)$ of a simplicial set $X$ as the $(n-1)$st homology of the (non-commutative) Moore complex $\tilde{G}(X)$, where $G_iX$ is ...

**6**

votes

**4**answers

473 views

### List of invariants that distinguish homotopy equivalent non-homeomorphic spaces

It is written on wikipedia article (https://en.wikipedia.org/wiki/Analytic_torsion) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy ...

**4**

votes

**0**answers

166 views

### Under what condition is a fiber bundle cobordant to the trivial bundle?

Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...

**3**

votes

**0**answers

163 views

### How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...

**3**

votes

**0**answers

65 views

### Various definitions of the odd Chern character form

I am asking this question from my possibly defected memory, so the things below may not be accurate.
I want to know how many different definitions of the odd Chern character form using differential ...

**3**

votes

**0**answers

149 views

### Topology on $\mathcal{C}(X,Y)$ to work with homotopy

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...

**2**

votes

**2**answers

213 views

### For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?

Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....

**5**

votes

**1**answer

243 views

### Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...

**12**

votes

**1**answer

136 views

### The finiteness criterium $F$ under quasi-isometry

A group $G$ is defined to have $F$ if there exists a finite $K(G,1)$.
This property is clearly not invariant under quasi-isometry as one can see from the trivial group and $\mathbb{Z}_2$.
My question:...

**1**

vote

**1**answer

190 views

### free group actions on a contractible topological space [closed]

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...

**9**

votes

**1**answer

298 views

### When do non-exact functors induce morphisms on $K$-theory?

Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...

**2**

votes

**1**answer

252 views

### Collapse of Hirzebruch Spectral sequence

This question is actually about reading Adams' Stable Homotopy and Generalised Cohomology; in Part II chapter 2, there are two numbered lemmata (Lemma 2.5 contravariant, 2.14 covariant) to the effect ...

**5**

votes

**2**answers

235 views

### When is $\mathbb C^d\setminus\mathcal Z$ simply connected?

Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho_1,\cdots,\rho_d\}$. Consider the co-ordinate ring $\mathbb C[x_1,\cdots,x_n]$. Let $\mathcal Z=\bigcup_C\mathcal V(x_i\ |...

**6**

votes

**0**answers

148 views

### Identifying a certain element of $E^{2,0}_2$ with the orientation class of an oriented spectra

I'm interested in filling the details of the well known computation, via AHSS of $E^*(\mathbb{C}P^n)$; where $(E,x_E)$ is an oriented spectrum.
It is supposed to be easy, yet every proof I've read ...

**16**

votes

**0**answers

505 views

### Is the determinant of cohomology a TQFT?

If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$...

**4**

votes

**1**answer

387 views

### “Polygons and gravitons” and Kodaira's theorem

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 470. At this point, he does some computations and obtains the conformal structure of the real ...

**6**

votes

**1**answer

309 views

### Bar/Cobar Adjunction Between Modules and Comodules

There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...

**8**

votes

**1**answer

273 views

### When is bar-cobar duality an equivalence?

Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...

**5**

votes

**2**answers

276 views

### Serre fibration vs Hurewicz fibration

What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?
Sorry if the question is too elementary for MO.

**2**

votes

**0**answers

71 views

### group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...

**4**

votes

**0**answers

147 views

### Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in:
$M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...

**3**

votes

**0**answers

94 views

### h-principle on Hilbert manifolds

Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...

**6**

votes

**0**answers

148 views

### Explicit diffeomorphism between an infinite dimensional sphere its product with itself

Let $S$ be an infinite dimensional sphere in a Hillbert space.
As $S$ is homotopic to the product $S \times S$, then $S$ is diffeomorphic to $S \times S$ (for Hilbert manifolds, a homotopy ...

**2**

votes

**1**answer

251 views

### induced group actions and covering maps on Eilenberg-Maclane space

Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma_k.
...

**6**

votes

**0**answers

97 views

### mod $p$ homology module of unordered configuration spaces of the projective plane

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...

**3**

votes

**1**answer

129 views

### Framed bordism class of the circle has order $2$

As the question title says, how do I see that the framed bordism class of the circle has order two?

**7**

votes

**0**answers

111 views

### Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices

Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...

**12**

votes

**1**answer

295 views

### Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...

**9**

votes

**2**answers

354 views

### Homotopy groups of Moore spaces

Is there anything known about the homotopy groups of the Moore spaces $M(\mathbb Z_m,n)$ if $m\neq 2$ and $n \geq 2$?

**1**

vote

**1**answer

238 views

### Free Symmetric Operads and $\mathbb{S}$-modules

In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads
have still an underlying free $\mathbb{S}$-...

**8**

votes

**0**answers

87 views

### Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces
Recall some definitions first:
Given a category $\mathcal{C}$ internal in $\...

**14**

votes

**1**answer

217 views

### Vector bundles with exactly one nonzero SW-class

I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...

**2**

votes

**1**answer

174 views

### The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...

**21**

votes

**2**answers

954 views

### The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ...

**2**

votes

**2**answers

294 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**13**

votes

**3**answers

544 views

### Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...

**2**

votes

**0**answers

93 views

### Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.

**13**

votes

**2**answers

497 views

### Integral cohomology of $G/N(T)$

Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^*(G/T,\mathbb{Q})$ is the regular ...

**7**

votes

**1**answer

226 views

### Thorough reference on regular homotopy

I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy ...