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

**5**

votes

**1**answer

195 views

### Homotopy type of embeddings of circle in the plane

What is the homotopy type of the space of (topological) emdeddings of $S^1$ in $\mathbb R^2$?
My conjecture: This space deformation retracts to $S^1\sqcup S^1$, and a retraction in each of ...

**-3**

votes

**1**answer

168 views

### Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...

**2**

votes

**0**answers

127 views

### Is the suspension of a weak equivalence again a weak equivalence?

Of course, the answer to this question depends on what we mean by suspension. If we work with based spaces and take the reduced suspension, the answer seems to be NO:
Take $X = \mathbb N$ (a ...

**2**

votes

**1**answer

114 views

### Unordered configuration space of $\mathbb{R}P^1$

In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): ...

**0**

votes

**0**answers

128 views

### Do there exist nontrivial motivic cohomology operations preserving weights?

Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...

**1**

vote

**1**answer

174 views

### Multiplicative structure in the cohomological Leray-Serre spectral sequence - please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set:
$X_p = \pi^{-1}(B^p)$,
$F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...

**1**

vote

**1**answer

167 views

### Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...

**8**

votes

**1**answer

185 views

### Contractible and Delta-generated implies strong deformation retract to a point?

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point.
However for general spaces it is well-known that just because a space is contractible, it does ...

**10**

votes

**1**answer

290 views

### Action of $\mathbb{CP}^\infty$ on $U(\infty)$

For a finite CW-complex $X$, the K-theory group $K^{-1}(X)$ is isomorphic to the group of homotopy classes of maps $[X, U(\infty)]$. The group of isomorphism classes of line bundles on $X$, which I ...

**0**

votes

**1**answer

269 views

### cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.
Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...

**3**

votes

**0**answers

136 views

### Reference request: Flipping the factors in the Künneth formula

I would like to know if there is a reference for the fact that the following diagram commutes:
$$
\begin{array}{ccccccccc}
0 & \to & H_*(X) \otimes H_*(Y) & \to & H_*(X\times Y) & ...

**5**

votes

**0**answers

95 views

### Can stable stems be generated by homotopy operations?

The motivation for this question comes from J. Cohen's result; at the prime $p=2$ his result says that any element in ${_2\pi_*^s}$ can be written as a (higher) Toda bracket of $2,\eta,\nu,\sigma$, ...

**0**

votes

**0**answers

202 views

### Serre Spectral Sequence and Cohomology Ring of Circle Bundles

I have the following (maybe simple) question about the cup product structure in the Serre spectral sequence.
Consider a fiber bundle $S^1 \rightarrow E \rightarrow B$ with euler class $e \in H^2(B)$. ...

**9**

votes

**1**answer

182 views

### Dyer-Lashof operations and the homology of GL_n

For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...

**3**

votes

**1**answer

107 views

### LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question:
Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...

**2**

votes

**2**answers

391 views

### Exact sequences of pointed sets - two definitions

It seems to me that there are (at least) two notions of exact sequences in a category:
1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite ...

**25**

votes

**1**answer

999 views

### Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...

**1**

vote

**1**answer

175 views

### Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...

**21**

votes

**2**answers

1k views

### Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...

**1**

vote

**0**answers

32 views

### Lattice-isotopic essentialization of arrangements

I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...

**2**

votes

**0**answers

127 views

### A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
...

**4**

votes

**1**answer

237 views

### When are principal bundles preserved by colimits?

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...

**2**

votes

**0**answers

68 views

### first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...

**5**

votes

**1**answer

233 views

### A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...

**4**

votes

**1**answer

235 views

### A question on Hawaiian earring

I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated.
Consider the Hawaiian earring. ...

**18**

votes

**1**answer

401 views

### Can topological cyclic homology compute Picard groups?

Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism
$$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$
where $Pic(\mathcal{O}_K)$ is the ...

**1**

vote

**0**answers

148 views

### Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...

**6**

votes

**0**answers

184 views

### Are Bökstedt's THH manuscripts available?

In many papers dealing with topological Hochschild homology, the original unpublished manuscripts by Bökstedt are cited. To name one example, in McClure and Staffeldt's On the topological Hochschild ...

**0**

votes

**0**answers

50 views

### Morphism of modules of sections and pullback bundles

I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) ...

**6**

votes

**2**answers

287 views

### Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...

**3**

votes

**2**answers

241 views

### Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...

**3**

votes

**1**answer

80 views

### Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$

Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
...

**4**

votes

**0**answers

148 views

### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

**-1**

votes

**4**answers

405 views

### Studying topology: which first, algebraic or differential? [closed]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**17**

votes

**1**answer

495 views

### What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$.
Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ ...

**2**

votes

**0**answers

151 views

### Fiber bundle in smooth category and topological category

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by ...

**4**

votes

**0**answers

118 views

### TQFT characterization of braiding statistics

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics.
Take $2+1$ dimensional Abelian Chern-Simons ...

**2**

votes

**1**answer

85 views

### Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white.
It is easy to find a set of generators for $B_{n,n}$:
$$
\begin{cases}
...

**3**

votes

**0**answers

177 views

### Using $\mathcal{U(H)}$ as a model for $EG$ and working with the Fredholm Operators

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on ...

**5**

votes

**1**answer

119 views

### Coverings/Cech cohomology of totally disconnected spaces

For any topological space $X$ we have a natural functor
$\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$
from the category of coverings of $X$ to the category of functors $\pi_1(X) ...

**3**

votes

**1**answer

164 views

### Parallelizable nearly-Kahler manifolds

In this question, we have discussed how the following bundle:
$E_{d} = TS^{d}\oplus \Lambda^2 T^{\ast}S^{d}$
is always trivial, where $S^{d}$ is the $d$-dimensional standard sphere. Now, let us take ...

**6**

votes

**0**answers

151 views

### Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a ...

**8**

votes

**2**answers

427 views

### Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group.
It is well-known that there is a well-defined map
$$
0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$
where ...

**6**

votes

**3**answers

310 views

### A conjecture about parallelizable generalized spheres

Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E_{d} = TS^{d}\oplus \Lambda ...

**2**

votes

**1**answer

188 views

### Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: ...

**5**

votes

**1**answer

180 views

### Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...

**4**

votes

**2**answers

287 views

### Maps to the group completion

Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary):
What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...

**6**

votes

**1**answer

220 views

### is this map a closed inclusion?

I apologize in advance if this question is too technical. I haven't found a reference in the literature yet, and it seems difficult enough that perhaps it has not been answered.
Let $A$, $B$, and $C$ ...

**0**

votes

**0**answers

79 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...

**0**

votes

**0**answers

82 views

### A suitable (transfer) map of Thom spectra $BT(n)^{-ad_{O(n)}|_{T(n)}-\gamma_1^{\times n}}\to BT(n)^{-\rho_n}$

For a Lie group $G$, write $ad_G=EG\times_G g\to BG$ for the adjoint bundle, $g$ is the Lie algebra of $G$ on which $G$ acts through its adjoint representation. Let $T(n)=O(1)^{\times n}$. I am ...