Homotopy, stable homotopy, homology and cohomology, homotopical algebra.
-2
votes
1answer
85 views
Degree of a rational function [on hold]
I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach):
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of ...
0
votes
0answers
61 views
Relating overlapping simplicial complexes
Let $X$ be a simplicial complex and let $A,B\subset X$ be
subcomplexes such that $C=A\cap B$ is a non-empty simplicial
complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$,
...
1
vote
1answer
87 views
configuration spaces of real projective space
Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any ...
2
votes
0answers
74 views
When does a map in the stable homotopy group gets killed when smashed with cone of itself?
Consider an element $e \in \pi_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence
$$ S^n \to S^0 \to C.$$
...
4
votes
1answer
182 views
Massey products and $A_{\infty}$ structures
I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A_{\infty}$ algebra. If my understanding is correct, the ...
4
votes
2answers
401 views
Based loop groups as stacks?
I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension ...
2
votes
1answer
164 views
Loop space structures on $RP^\infty$
I am interested in infinite loop structures on the infinite dimensional projective space $\mathbb{R} P^\infty$. Is it unique? I think this has to be known in work of May, and If so, then I presume its ...
4
votes
1answer
139 views
liftings of principal bundles
I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise.
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...
4
votes
1answer
296 views
Boardman-Vogt tensor product
Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category ...
14
votes
1answer
378 views
Free Loop-Space Recognition Principle
It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...
10
votes
1answer
330 views
Coboundary of a cup-product
Let $A\subset X$ be CW-complexes (or even manifolds). In cohomology with coefficients in a commutative ring $R$, we have a long exact sequence
$$\cdots \rightarrow H^p(X,A)\rightarrow ...
3
votes
2answers
200 views
Weak homotopy equivalence and Cech cohomology
If two topological spaces are weak homotopy equivalent to each other, are their Cech cohomology groups the same?
1
vote
0answers
43 views
What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$? [migrated]
What is the quotient space $\mathbb{T}^3/\mathbb{Z}_2$, When $\mathbb{Z}_2$ acting on 3-dimensional torus $\mathbb{T}^3 := \mathbb{R}^3/\mathbb{Z}^3$ by sending $x$ to $-x$? Does anyone know how to ...
2
votes
0answers
112 views
A Cartesian model structure (and straightening for) on $n$-trivial simplicial sets
A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a ...
17
votes
1answer
386 views
Super-cobordisms
One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
2
votes
2answers
169 views
A line bundle over the manifold of singular matrices
According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we ...
1
vote
0answers
163 views
Intuitive Approach to Sheaf and Cech Cohomology [closed]
Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
8
votes
0answers
307 views
Pairing of cohomology and homology Künneth formulas
Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian ...
9
votes
2answers
626 views
Symplectic K-theory
For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...
-3
votes
0answers
90 views
cohomology of permutation group of order power of $2$
Let $S_k$ be symmetric group of order $k$. What is
$$
H^*(BS_{2^k};\mathbb{Z}_2)?
$$
$$
H^*(BS_{2^k};\mathbb{Z})?
$$
I only know the case $k=1$.
0
votes
0answers
51 views
discrete group action on Stiefel manifold
Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by
$$
(1,2)(u,v)=(-u,v-u),
$$
$$
...
20
votes
1answer
642 views
fake $S^{2k}\times S^{2k}$
Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...
3
votes
0answers
173 views
What are iterated cobar constructions?
In Beck's paper "On H-spaces and Infinite Loop Spaces", he states that every algebra over the monad $\Omega^k$$\Sigma^k$ is a $k$-fold loop space. He proves the trivial case k = 0 when this is the ...
2
votes
1answer
125 views
Fibration $p : \tilde Y \to Y$ with discrete fiber induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$
If $X$ is simply connected, locally path connected space and $p : \tilde Y \to Y$ is a covering map then it is easy to show that it induces bijection $p_*:[X, \tilde Y]_* \to [X, Y]_*$. Let's weak ...
2
votes
1answer
111 views
cohomology of orthogonal (or general linear) group over finite fields
Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
...
4
votes
3answers
284 views
Need examples of homotopy orbit and fixed points
I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or ...
9
votes
2answers
180 views
Maps with Hopf invariant zero are suspensions
Let $h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi_{2n-1}(S^n)$ satisfies ...
5
votes
1answer
160 views
Symmetric L-groups of integral group ring of finite cyclic groups
Where can i find the results about $L^{\ast}(\mathbb{Z}\pi)$ for $\pi$ a finite cyclic group?
12
votes
3answers
654 views
Homology equivalence and isomorphism on $\pi_1$ not enough for homotopy equivalence?
It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).
...
11
votes
1answer
227 views
What is obstructing two stably-isomorphic vector bundles from being isomorphic?
The specific situation is the following:
Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...
-1
votes
0answers
42 views
Universal covering space of CW complex has CW complex structure [migrated]
We know that covering spaces of many low dimensional CW complexes such as graph (CW structures with only 0-cells and 1-cells) and all compact surfaces has CW structure. [The second fact is due to ...
4
votes
1answer
214 views
Obstruction to a $SU(4)$-structure in eight dimensions
What is the obstruction for the existence of a $SU(4)$-structure on a spin, eight-dimensional manifold $M$? This is equivalent to the existence of two nowhere vanishing global sections of the ...
17
votes
3answers
556 views
Center of a simply-connected simple compact Lie group and McKay correspondence
Let $G$ be a simply-connected simple compact Lie group. Its center $Z(G)$ is a finite abelian group, say $Z(G) = \mathbb Z/k\mathbb Z$ for $G=SU(k)$.
I find the following interpretation of $Z(G)$ in ...
3
votes
1answer
264 views
Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?
Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...
5
votes
0answers
114 views
Actions of cofibrations and induced maps of cofibres
Working in some nice category of based topological spaces (compactly generated with CW homotopy type, say) suppose we have a homotopy commutative diagram
$$
\begin{array}{ccccc}
& & j & ...
1
vote
1answer
166 views
cohomology of orthogonal group of integers
Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...
1
vote
1answer
286 views
cohomology ring of symmetric group of order $3$
Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring
$$
H^*(S_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S_n$.
For $n=1$, $S_1$ is trivial. ...
0
votes
1answer
242 views
History of Poincare conjecture in higher dimension [closed]
As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to ...
2
votes
0answers
142 views
easy question involving definition of bousfield localization [migrated]
Easy question: If $X$ is a space/spectrum such that $H_*(X)=0$ (where $H_*$ is ordinary homology), then why does this imply the Bousfield localization $L_HX$ is contractible?
My attempt: $L_HX$ is ...
-3
votes
1answer
107 views
partial converse of existence of covering spaces [closed]
Suppose $X_1$ and $X_2$ are two spaces which has a common finite sheeted covering space $Z$ (may be distinct index)...from here can we conclude that there exists a space $W$ such that $X_1$ and $ X_2$ ...
-1
votes
0answers
146 views
Flat vector bundles and constant transition functions [migrated]
Let $E\to M$ be a vector bundle endowed with a flat connection. Then, does $E$ admit a bundle atlas with constant transition functions?
For a vector bundle with constant transition functions, are ...
6
votes
1answer
432 views
Homologically distinct infinite loop structures on a space
Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...
1
vote
0answers
101 views
Spectral sequences and Batalin-Vilkovisky formalism
I have been studying the BRST quantization in quantum field theory recently and noticed that the subject is very much related to algebraic topology and cohomology. A quick google search led me to the ...
3
votes
1answer
159 views
Centralizers in the universal central extensions of the alternating groups?
For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
2
votes
0answers
97 views
Is the Thom diagonal co-$E_\infty$?
Given a map of spaces $f:X\to BGL_1(R)$ for $R$ an $E_\infty$-ring spectrum (of course this can be done more generally) one can produce a Thom spectrum $Mf$ by a number of methods. Let's denote such a ...
2
votes
1answer
130 views
the Pontryagin number of a 4-dim orientable surface bundle with fiber of genus 2
Is the following statement correct or known to be correct?
For a 4-dimensional closed orientable
surface bundle $E$ with fiber of genus 2, the signature must be 0 mod 8 (or the Pontryagin number of ...
7
votes
0answers
146 views
Unicity of Johnson-Wilson Theories
Let $E$ be a ring spectrum with a $p$-typical complex orientation. Then we call $E$ a form of $BP\langle n\rangle$ or a generalized $BP\langle n\rangle$ if the induced map
...
0
votes
0answers
117 views
Pontryagin class of quaternionic line bundle
Let $\xi^{\mathbb{C}}$ be a complex line bundle over a CW complex $B$. Then
$$
VB_{\mathbb{C}^1}(B)\cong [B,BU(1)]=[B,\mathbb{C}P^\infty]=[B,K(\mathbb{Z},2)]\cong H^2(B;\mathbb{Z}).
$$
Hence if ...
-3
votes
1answer
141 views
Fibre bundles and flat connections [closed]
If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat ...
2
votes
1answer
129 views
Two-bridge knots and CW-complex
The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation.
On the other hand, it's possible to provide a CW-complex with only one 0-cell ...
