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

**4**

votes

**0**answers

41 views

### What is know 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 ...

**2**

votes

**0**answers

48 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 $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...

**6**

votes

**0**answers

65 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

237 views

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

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

47 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

506 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

64 views

### topology- Compute π1(M3) when M3 be the 3-manifold [on hold]

Let M3 be the 3-manifold obtained by gluing two handlebodies of genus g by
the identity map. To be precise: Let H1, H2 be two copies of a “standard” genus
g handlebody in S
3
. If we denote Σ1 = ∂H1, ...

**2**

votes

**0**answers

80 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 ...

**9**

votes

**0**answers

105 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

56 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 ...

**4**

votes

**0**answers

75 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

157 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

63 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 ...

**5**

votes

**4**answers

444 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 ...

**3**

votes

**0**answers

132 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

153 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

60 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

143 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

205 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

186 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 ...

**-1**

votes

**0**answers

121 views

### Homotopy equivalence and sheaf cohomology

I have an inclusion $Y \hookrightarrow X$ of varieties that is a homotopy equivalence. ($X$ is a toric variety, $Y$ is a hypertoric variety, in case that is important)
I know $H^0(\mathscr{O}_X)$, ...

**11**

votes

**1**answer

130 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

186 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

263 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

239 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

233 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

147 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

465 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 $D(M) \to D(N)$...

**3**

votes

**1**answer

294 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

278 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

245 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

263 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

70 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

93 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

247 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

94 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

128 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?

**6**

votes

**0**answers

98 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

288 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

338 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

236 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

86 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

206 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

170 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

920 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

287 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

537 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 ...

**1**

vote

**0**answers

87 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.