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

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58 views

Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
7
votes
0answers
87 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
2answers
281 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
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1answer
165 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 underling free ...
0
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0answers
44 views

How to compute a signature of n-dimensional torus [on hold]

I can use cup product on cohomology, but what exactly means fundamental class? Should I use Chern-Weil correspondence to compute its Pontrjagin class (actually what does Pontrjagin class for ...
8
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0answers
83 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 ...
13
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1answer
171 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
1answer
159 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 ...
19
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2answers
823 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
2answers
252 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 ...
12
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3answers
473 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
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0answers
82 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.
13
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2answers
473 views

Integral cohomology of $G/N(T)$

Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^*(G/T,\mathbb{Q})$ is the regular ...
7
votes
1answer
217 views

Thorough reference on regular homotopy

I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy ...
5
votes
1answer
327 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
1
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0answers
64 views

Trefoil Knot Seifert Minimal Surface Equation

I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following: Starting from a real ...
1
vote
1answer
277 views

Torsion of $H_{n-1}$

Suppose $X$ is a non-orientable manifold. Using Universal Coefficient Theorem (UCT) for homology, we can get that the torsion of $H_{n-1}$ is a cyclic group of order $2$. I am looking for a proof of ...
14
votes
1answer
296 views

Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
9
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0answers
269 views

Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions $G$ and $H$ are finite groups and $K$ an infinite group. there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...
1
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1answer
176 views

Deformation Quantization

I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start?
3
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1answer
206 views

Poincaré Duality for non-compact manifolds without Zorn's Lemma

Does exists a proof of the Poincaré Duality version for non-compact manifolds without using the Zorn's Lemma? I know that there is a proof using the Whitney embedding theorem, but I don't know this ...
6
votes
2answers
216 views

Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal ...
0
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0answers
87 views

de Rham type cohomology for covariant derivative?

We know that in general for a covariant derivative $D$ on a vector bundle $\xi$ over $M$ we don't have $D \circ D = 0$. This prevents us from having the following cochain complex \begin{equation} ...
0
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1answer
141 views

Atiyah-sequence-like definition of connection on vector bundles?

For a principal bundle $\pi: P \to M$ we have the following Atiyah sequence that can be used to define a connection on it \begin{equation} 0 \to V{P} \to T{P} \to \pi^*{T{M}} \to 0 \end{equation} A ...
0
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0answers
152 views

Every vector bundle can be induced from a principal bundle? its frame bundle? [migrated]

If it is a theorem could somebody tell me the name? If it is wrong could somebody give a counterexample to illustrate what the obstruction is? I am wondering this because in Clifford Taubes' book on ...
10
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2answers
463 views

Reference request: Goodwillie tower of the identity

The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor $$ X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]}) $$ in ...
15
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0answers
259 views

Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
8
votes
2answers
339 views

Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids

Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations. Generally $Vect(BG) \ne ...
7
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0answers
103 views

Reference request: Atiyah-Segal completion on spectrum level

It seems like the Atiyah Segal completion theorem for the two element group $G = \mathbb Z_2$ and one-point space $X=\{ * \}$ with trivial G action yields a statement about the underlying spectra as ...
5
votes
1answer
292 views

Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
4
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0answers
140 views

Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
2
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0answers
198 views

E-infinity operads explicit examples

I was looking for particular and explicit examples of $E_\infty$-operads. I know the $E_\infty$-operad defined by Smith in http://arxiv.org/abs/math/0004003, and the Barratt-Eccles operad, but it is ...
13
votes
1answer
464 views

$p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
1
vote
1answer
205 views

Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in Math Stack Exchange. I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
5
votes
0answers
95 views

Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle). I ...
0
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0answers
31 views

Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
6
votes
0answers
101 views

Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2. I was wondering if there exists any reference concerning the ...
6
votes
1answer
177 views

Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
4
votes
1answer
130 views

“Small” simplicial complex with torsion trees

I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...
8
votes
0answers
161 views

Are there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any ...
5
votes
1answer
146 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...
9
votes
2answers
336 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
7
votes
1answer
412 views

what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$. What is G'? I know there is concrete description in terms of pairs ...
4
votes
2answers
194 views

Equivalence of different cohomology groups

Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups: (1) singular cohomology ...
3
votes
2answers
236 views

CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...
1
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1answer
172 views

Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets. I mean cases where adding a topology to the sets ...
5
votes
1answer
74 views

smoothing locally-finite (Borel-Moore chains)

Let $M$ be a smooth manifold. As is recorded in (for example) Lee's book, de Rham proved that one can calculated singular homology, $H_*(M)$ using smooth simplices. Does the result extend to ...
2
votes
1answer
238 views

Local “pathologies” in spaces arising naturally in algebraic topology

I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover ...
6
votes
1answer
242 views

“structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration". Does it make sense to talk about "structure ...
2
votes
1answer
231 views

Using Lefschetz duality in algebraic geometry

I am reading the paper of Fulton and Lazarsfeld on the connectivity of degeneracy loci of morphisms of vector bundles, but there is a comment in the article that I don't quite understand. Let $G$ be ...