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

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12
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235 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 ...
0
votes
0answers
17 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
0answers
100 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
1answer
163 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
0answers
60 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
1answer
190 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
1answer
221 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. ...
-1
votes
0answers
60 views

Lemma: (Path -Cantor Lifting)? [on hold]

Definition: Let $p:E\rightarrow B$ be a map. If $f:X\rightarrow B$is a map, a lifting of is a map $\widetilde{f}:X\rightarrow E$ such that $p\circ \widetilde{f}=f$ ¿TRUE or FALSE? "Let $C$= Cantor ...
18
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1answer
348 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
votes
0answers
25 views

Geodesic parameterization under conformal mapping [closed]

Under a conformal deformation of the euclidean metric, say: $\hat{g}_{ij}=e^{\phi}\delta_{ij}$, where $\phi$ depends on the radial coordinate alone, I am struggling to see the following fact: "With ...
1
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0answers
139 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 ...
-1
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0answers
81 views

Is there a version of decomposition theorem for smooth open manifolds in dimensions grater than 3? [closed]

Is there a version of decomposition theorem http://www.jstor.org/stable/2034963 for smooth open manifolds in dimensions grater than 3? There is two things about the cited theorem: it is for ...
5
votes
0answers
163 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
0answers
46 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) ...
0
votes
0answers
37 views

Betti numbers over unital rings [closed]

Is the following statement correct? Given a manifold M. If H_1(M,Z) is a finite cyclic group, then the first R-betti number b_1(M,R) is bounded from above by 1 for every unital ring R.
6
votes
2answers
261 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
2answers
223 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
1answer
75 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: ...
2
votes
0answers
109 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 ...
-2
votes
4answers
351 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 ...
16
votes
1answer
355 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
0answers
145 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
0answers
102 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
1answer
78 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
0answers
175 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
1answer
108 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
1answer
157 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
0answers
146 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 ...
6
votes
2answers
296 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
3answers
297 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
1answer
180 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
1answer
169 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
2answers
275 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
1answer
214 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
0answers
76 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
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0answers
78 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 ...
10
votes
1answer
264 views

Cohomology of the Image of J spectrum

Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e. $$ [j, HZ/p]_*$$ as a module over Steenrod algebra. ...
4
votes
0answers
103 views

Are there any known ``topological" invariants for branched coverings?

My question is the following: let $f:\Omega\to \mathbb{R}^n$ be a branched covering, namely $f$ is continuous, discrete (each fiber is a discrete subset of $\Omega$) and open (open sets get mapped ...
7
votes
1answer
222 views

What is an example of a formal group law that is Landweber-exact but not flat?

Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v_1x^{p^1} + ... + v_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact ...
1
vote
1answer
117 views

Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold. Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$. Let $S_3$ be the symmetric group of order $3$. Let $S_3$ act on $F(M,3)$ by ...
8
votes
1answer
248 views

When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...
3
votes
0answers
73 views

Reference request: linearly independent cycles in a manifold

The following seems to be well known to experts, but I would be happy if there is a paper or textbook that I can cite. Note: all of the manifolds are assumed to be without boundary. Suppose that $C$ ...
0
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0answers
46 views

Product structure on manifolds via lifting classifying maps

Let's say you want to study $d$-dimensional manifolds $M$ which decompose functorially into $M\cong N\times P$ for a fixed $P$. Can this structure be expressed by a lift of the stable normal bundle? ...
9
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0answers
197 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
3
votes
1answer
158 views

How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?

It is often said [e.g. Atiyah, "Bordism and Cobordism" (1961)] that the Thom spectrum $MSO(i)$ represents oriented cobordism, in the following sense: \begin{eqnarray} MSO^n(X,Y) &:=& \lim_{i ...
2
votes
0answers
173 views

Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid. Are ...
5
votes
1answer
165 views

Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following. There exist $\Sigma$-free operads ...
2
votes
0answers
114 views

Pro-p topology on free group

Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
1
vote
1answer
157 views

Homotopy type of certain maps on complex grassmanian

$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid ...
1
vote
0answers
87 views

Explicit calculation of G-CW(V) structure of a G-space

I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...