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

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-4
votes
4answers
209 views

Studying topology: which first, algebraic or differential? [on hold]

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 ...
9
votes
0answers
138 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
120 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
84 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
68 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} ...
-5
votes
0answers
57 views

vector bundle and characteristic classes [on hold]

why every orient able line bundle is trivial? I give an orientation s(x) to each fib re such that s(x).s(x)=1 using the euclidean metric but I can not understand why this section becomes continuous
3
votes
0answers
123 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
102 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) ...
2
votes
0answers
68 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 ...
5
votes
0answers
125 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 ...
5
votes
2answers
269 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 ...
5
votes
3answers
265 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
173 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
162 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
268 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 ...
-3
votes
0answers
112 views

Does the diffeomorphism group of an open manifold act naturally on the compactification of the manifold? [closed]

I need help regarding this question please: Does the diffeomorphism group of an open manifold act trnsitively/freely on the compactification of the manifold? any recommended references!
6
votes
1answer
206 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
75 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
0answers
74 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
251 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
101 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
212 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
116 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
242 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
72 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
votes
0answers
41 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
votes
0answers
193 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
148 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
133 views

Finitness 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
150 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
109 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
156 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
84 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 ...
5
votes
2answers
289 views

Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes

I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$. Milnor's On spaces having the homotopy type ...
1
vote
1answer
120 views

Cofiber sequence $A\vee A \to A \wedge A \to \bar{A}\wedge \bar{A}$ for a spectrum $A$

For concreteness, let us work with the language of spectra introduced in EKMM. In Strickland's paper "Products on $MU$-modules", he proves the following. If $R$ is a q-cofibrant commutative ...
4
votes
0answers
230 views

Why does this fundamental group not have elements of finite order? [duplicate]

Let $X$ be a subset of $\mathbb R^3$ with its induced topology and let $a\in X$ be a point. Then the fundamental group $\pi_1(X,a)$ seems not to have elements of finite order (except the identity of ...
0
votes
0answers
78 views

conjugate operation on vector bundle

Is the conjugate operation on $\overset{\sim}{K}(\mathbb{C}\mathbb{P}^n)$ known? If so, can I get the full formula at least in terms of the basis $\eta^i$? Here $\overset{\sim}{K}(X)$ denotes the ...
1
vote
0answers
28 views

Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR

Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$? ...
4
votes
1answer
143 views

The image of the Hurewicz map for rational loop spaces

Let $K$ be the rationalization of a simply-connected finite CW complex. Then the Samelson product gives $\pi_*(\Omega K)$ the structure of a graded Lie algebra, and the Hurewicz map $h: \pi_*(\Omega ...
20
votes
0answers
299 views

Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
1
vote
1answer
222 views

Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$. We have the ...
0
votes
0answers
69 views

rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...
13
votes
2answers
384 views

What's an example of 2 elliptic curves with the same ground ring s.t. their associated cohomology theories detect different things?

My understanding is that a complex-oriented spectrum is a ring spectrum $E$ with a map $MU \to E$. Analogously, a ring with a formal group law is a ring $R$ with a map $L \to R$ (where $L$ is the ...
2
votes
1answer
128 views

Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered. Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...
0
votes
0answers
7 views

dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients. Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$. Let the dual basis be ...
24
votes
2answers
715 views

Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...
3
votes
1answer
219 views

Classifying spaces of topological groups whose underlying spaces are homotopy equivalent

Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...
8
votes
1answer
301 views

Must we know $MU^*(X)$ in order to compute $Ell^*(X)$?

Let $Ell^*(X)$ be the elliptic cohomology theory (associated to a given elliptic curve $E$) of a nice space $X$. Recall the Landweber-Ravenel-Stong construction: $MU^*(X) \otimes_{MU^*} R \simeq ...
3
votes
1answer
152 views

Toral rank conjecture

In 1985, S.~Halperin conjectured in the topological context of maximal free torus actions on topological manifolds, that: If $X$ is a topological space, then $$\dim H^*(X;\mathbb Q)\geq 2^{rk(X)}.$$ ...
1
vote
3answers
251 views

orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$. I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates. For ...