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

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2
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86 views

order of elements in a mapping space

Let $B$ be a finite CW-complex and $\xi$ be a vector bundle over $B$ with structure group $\Sigma_n$, the $n$-th symmetric group. Then corresponding to $\xi$, we have a classifying map $$ g\in \tilde ...
3
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2answers
126 views

equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
3
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1answer
179 views

Homology of solvable Lie groups made discrete

In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology. It is well-known how to compute the homology of abelian groups, ...
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0answers
88 views

Augmentation of the sphere spectrum

I am wondering if it is sensible to talk about the augmentation ideal of the sphere spectrum in the category of spectra, as well as the `submodule of decomposables', whose construction comes from the ...
3
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0answers
118 views

Topologized category of bounded chain complex

I am reading Segal's paper 'categories and cohomology theories' [1], but there is one claim (in the last example in sec.2) I don't quite understand: Let $\mathcal{C}$ be the category of bounded chain ...
5
votes
2answers
294 views

3-manifolds homotopy equivalent to a surface

I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S_g$ must itself already be homemorphic to $S_g \times ...
3
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1answer
201 views

the “Kahn-Priddy map” and “multiplicative $p$-local equivalence”

The following is a part of a paper that I need to understand I totally do not know the argument. Could you explain? Thanks. Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the ...
4
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0answers
147 views

Cofiltered diagram of path connected spaces with empty homotopy limit?

Is it possible to have a filtered category $J$, a functor $F: J^\mathrm{op} \to \mathrm{Spaces}$ such that $F(i)$ is path connected for all $i$ and such that $\mathrm{holim} F = \emptyset$? If $J$ is ...
4
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1answer
219 views

Triviality of a fiber bundle

Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?
0
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118 views

when is “fibering” preserved under homotopy equivalence

Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
1
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0answers
84 views

Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
1
vote
1answer
180 views

Different model structures on Top

There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure. In the Mixed model structure ...
1
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0answers
144 views

Homotopy equivalence of Lens spaces

I find the following statement about the homotopy equivalence of Lens spaces in Wikipedia. The three-dimensional spaces $L(p,q_1)$ and $L(p,q_2)$ are homotopy equivalent if and only if $q_1 q_2\equiv ...
12
votes
1answer
432 views

Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)

If $A$ is an abelian group, we have $Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$ where the left hand side is the space of self-homotopy equivalences. Is there an easy way to ...
4
votes
2answers
292 views

Gluing two 3 manifolds along their boundary

Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism ...
0
votes
1answer
147 views

Equivalence relations of topological spaces not comparable with homotopy [closed]

The question is pretty much contained in the title: What are examples of equivalence relations of topological spaces which are neither stronger nor weaker than homotopy equivalence? Something that ...
3
votes
0answers
167 views

Correspondence between Serre and Tate on resolution of singularities

I am told that there is a 1961 correspondence between J-P. Serre and J. Tate on resolution of singularities in characteristic 0, where Serre discusses his attempts to disprove the latter using R. ...
3
votes
0answers
98 views

Symmetric sub-simplicial sets of the nerve of $\mathbb{N}$

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$ degeneracies: inserting $0$ faces: adding consecutive entries ...
2
votes
2answers
238 views

Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now ...
4
votes
1answer
180 views

Whitney sum formula for Pontryagin classes II

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the ...
8
votes
1answer
250 views

Whitney sum formula for Pontryagin classes I

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this ...
6
votes
1answer
159 views

homotopy groups of an orbifold

The isometry group of the 3-dimensional hyperbolic space $\mathbb{H}^{3}$ is $PSL(2,\mathbf{C})$. What are the homotopy groups of the quotient space $\mathbb{H}^{3}/PSL(2,\mathbf{Z})$ ?
3
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1answer
182 views

Status of Zeeman's collapsability Conjecture

Zeeman's conjecture in topological combinatorics states that if K is a contractible polyhedron of dimension 2, then K×I has a collapsible subdivision. What is the status of this conjecture ...
14
votes
1answer
277 views

Multiplicative cohomology theories and smash products

In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says However, G. W. ...
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77 views

When closed subsets have finitely many connected componenets

Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?
12
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1answer
208 views

Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories

Given a finite subgroup of $G$ sitting inside the Morava stabilizer group $S_n$, we can form the homotopy fixed point spectrum $E_n^{hG}$. There is a spectral sequence with $E_2^{s,t} = ...
20
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1answer
529 views

How does Poincare duality interact with the Serre spectral sequence?

Suppose $F^m \to E^{m + n} \to B^n$ is a fiber bundle of closed oriented manifolds. I'm interested in understanding how the Serre spectral sequences for homology and cohomology of $E$ interact with ...
14
votes
3answers
863 views

Motivation for equivariant homotopy theory?

I'm in the process of learning equivariant homotopy theory, so I was wondering: what is the importance of equivariant homotopy theory, and what has it been applied to so far? I know of HHR's solution ...
10
votes
1answer
312 views

T-equivariant cohomology of flag variety

Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ...
7
votes
0answers
377 views

Algebraic geometry introduction for homotopy theorists/algebraic topologists

Algebraic geometry has a plenty of decent introductory texts now. Some are of the classical commutative algebraic approach(following EGA), like Ravi Vakil's "Foundations". Some use facts from ...
8
votes
2answers
791 views

Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?

In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle ...
4
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1answer
220 views

A question on complex line bundle over $S^{2}$

Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$. Assume that $\ell$ is a sub line bundle of ...
1
vote
1answer
116 views

triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle

Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle $$ \xi: ...
2
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0answers
109 views

Divisible fundamental group [duplicate]

I apologize if this question seems trivial or elementary. Is there any concrete topological space with divisible fundamental group? For example, is there any such a space the fundamental group in ...
5
votes
1answer
219 views

Graded Hopf algebras and H-spaces

Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both ...
3
votes
1answer
247 views

Stratification of complex algebraic varieties

Let $V$ be a complex quasi-projective variety, we know from H. Whitney's and B Teissier works on stratifications of algebraic varieties that $V$ has an intrinsic stratification $$X_0\subset ...
3
votes
1answer
187 views

geometric conditions on maps between manifolds inducing monomorphisms on cohomology

Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What geometric conditions on $f$ can we impose such that the induced homomorphism $$ f^*: ...
19
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0answers
258 views

Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
12
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1answer
420 views

Realizing symmetric groups by diffeomorphisms

Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...
2
votes
0answers
96 views

obstructions to embeddings of manifolds into Grassmannians

Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in ...
6
votes
1answer
145 views

Change of groups for naive G-spectra

Let $H$ be a subgroup of $G$ a compact Lie group and let $\text{spectra}[G]$ be the category of naive $G$-spectra (ie G-objects in the category of spectra). Then there is a forgetful functor $i^*$ ...
2
votes
1answer
189 views

Is this affine-subspace analogue of a Grassmannian a classifying space?

Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the ...
4
votes
3answers
192 views

A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural ...
8
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2answers
268 views

Groups with trivial rational homology and their finite index subgroups

For a short exact sequence $0 \to G \to H \to K \to 0$ of (discrete) groups with $K$ finite we have, as a consequence of the Hochschild-Serre spectral sequence, that $H^{\ast}(H;\mathbb Q) = ...
7
votes
0answers
168 views

When do contributions of $\pi_*(L_{K(n)}S^0)$ to $\pi_*(S^0)$ stabilize?

I apologize if this question is obvious as I am just beginning to learn this field. I am also abusing the dependance on $p$ for notational ease. The Chromatic Convergence theorem establishes that ...
7
votes
4answers
654 views

Is an entire function, with nowhere vanishing derivative, always a covering map?

Assume that $f:\mathbb C\to\mathbb C$ is entire, and also that $f'(z)\ne 0$, for all $z\in\mathbb C$. Does that imply that $f$ is a covering map of $f[\mathbb C]$? Clearly, $f$ is a local ...
10
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2answers
373 views

In CGWH, is every cofibration an inclusion with closed image?

As the title suggests, in CGWH, is every cofibration an inclusion with closed image?
0
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0answers
116 views

Homology of $S^n/G_x$

I tried to find the homology groups of the quotient of the unit sphere $S^{n-1}$ by an action of a finite subgroup $G$ of $SO(n)$. I'm especially concerned with $$H_i(S^{n-1}/G),\quad 1\leq i\leq ...
9
votes
4answers
419 views

Vietoris-Begle theorem for simplicial sets

I've learned the theorem when reading a comment by Vidit Nanda to my question see here. Here is the (simplified) version of the theorem for topological spaces: Vietoris-Begle Theorem Let ...
20
votes
1answer
548 views

Is the moduli of formal groups smooth?

There is a notion of smooth stack in "homotopical algebraic geometry 2" and a notion of a cotangent complex for certain kinds of stacks (including representable stacks). I have two questions. Is ...