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

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3
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2answers
105 views

Relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? $\pi_*(X, A)$ and $\pi_*(Y \cup_f X, Y)$?

Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let $f: A \to Y$ be a cellular map. What is the relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? Is there a similar ...
1
vote
0answers
11 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
1
vote
1answer
75 views

Parametrized Atiyah-Singer index theorem

Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D_s:E\to F,s\in S$ be a continuous family ...
1
vote
0answers
56 views

Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$ [migrated]

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...
20
votes
2answers
692 views

What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
2
votes
1answer
57 views

Equivariant Tubular Neighborhood Theorem in Hilbert Spaces

I'm looking for a proof of the above theorem in the case of a compact Lie Group $G$ acting on an (infinite dimensional) Hilbert Space $\mathbb{H}$ (Compare T. Dieck "Transformation Groups", Theorem ...
6
votes
0answers
183 views
+50

Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
3
votes
0answers
39 views

Adjunctions of uniformly locally connected spaces

A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and ...
7
votes
1answer
103 views

cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra (1) $$ H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p) $$ for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...
12
votes
1answer
307 views

Integral cohomology ring of K(Z,3)

Computing the cohomology of Eilenberg Maclane spaces is a feasible but difficult problem in algebraic topology. The general answer is quite complicated (see the MO answer and the reference therein ...
35
votes
1answer
801 views

Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
0
votes
1answer
220 views

Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...
3
votes
1answer
225 views

coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...
5
votes
2answers
222 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
4
votes
1answer
123 views

The evaluation fibration of a transitive, effective topological group action

Does anybody know a reference to the following fact? If $G$ is a topological group acting transitively and effectively on a space $X$, then the evaluation map $G \rightarrow X$, $g \mapsto g \cdot ...
12
votes
2answers
448 views

Classifying spaces for enriched categories

Is there a standard construction of a classifying space $BC$ for a category $C$ which is enriched which takes into account the enrichment? This is of course vague... The simplest example I can think ...
3
votes
1answer
173 views

The structure map of topological K-theory

This may be a silly question but I don't know the answer. I know the construction of (equivariant) K-spectrum $KU_G$ and the periodicity of (equivariant) K-theory. But I don't know its structure maps ...
12
votes
1answer
212 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
3
votes
1answer
107 views

Classifying space for homology endomorphisms supported on a graph?

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the ...
5
votes
1answer
383 views

Is the face poset a Heyting algebra?

Is the face poset of a simplicial or nice enough cellular complex a Heyting algebra in some natural way? Edited to add: For the benefit of illustration, here's a few face posets: the boundary of a ...
0
votes
0answers
71 views

A simplicial complex which is collapsible but there exist a subdivision of it does not [closed]

Does anyone know a simplicial complex which is collapsible but there exist a subdivision of it which does not?
9
votes
1answer
265 views

Contractible topological groups

Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?
0
votes
0answers
91 views

$G$-CW complex structure of certain G-space

Let $G$ be a finite group and $H$ be a subgroup of $G$ . Let us denote $X= G/H \ast G/H \ast \cdots \ast G/H $($k$ times).Where $\ast$ denotes the topological join operation. My question are as ...
14
votes
3answers
521 views

Homology generated by lifts of simple curves

Let $\Sigma$ be a compact connected oriented surface and $p:\tilde{\Sigma}\to\Sigma$ a finite regular cover. Consider the set $\Gamma$ of simple closed curves on $\tilde{\Sigma}$ obtained as a ...
32
votes
2answers
2k views

Homotopy groups of $S^2$

in the paper Foundations of the theory of bounded cohomology, by N.V. Ivanov, the author considers the complex of bounded singular cochains on a simply connected CW-complex $X$, and constructs a ...
0
votes
0answers
123 views

p-local space vs p-completion.

I have some trouble to understand the difference between the p-completion and p-local space. if $X$ a simply connected spaces such that all higher homotopy groups are finitely generated groups then ...
6
votes
0answers
147 views

Intuition behind the following theorem of Reeb?

What is the intuition behind the following theorem of Reeb? If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.
17
votes
3answers
949 views

Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
4
votes
0answers
93 views

No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...
6
votes
1answer
212 views

Differential geometry without the Hausdorff condition or the second axiom of countability

I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things ...
5
votes
1answer
177 views

group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
6
votes
1answer
179 views

How does one identify flow lines on a vector bundle with those on the base in Morse theory?

In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function ...
5
votes
1answer
253 views

Cohomology of $G_3(\mathbb{R}^5)$

This is in some sense a specialization of the question integral or rational cohomology of real grassmannians. Let $G_3(\mathbb{R}^5)$ denote the real Grassmannian of (unoriented) $3$-planes in ...
1
vote
1answer
190 views

realization map for K-theory of spheres

Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that ...
4
votes
0answers
129 views

When is a circle fibration a circle bundle?

Let $\pi : E \to B$ be a Serre fibration over a CW complex, with circle fibers. In the orientable case, it is easy to see that $\pi$ is fiber homotopy equivalent to a principal $SO(2)$--bundle. ...
2
votes
0answers
64 views

Quotients of simplicial complexes which are simplicial complexes

In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...
5
votes
1answer
372 views

Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?

I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me). The cited question/answer ...
5
votes
0answers
95 views

Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...
0
votes
0answers
48 views

Characterization of Singular locus

Let A be a complete regular local ring over a field k and B be a complete normal local ring over a field k. We assume that (Krull-dimension of A) > 1. We consider the ring homomorphism f: A ---> B, ...
5
votes
1answer
185 views

can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the cohomology of some Eilenberg-Maclane space $K(\pi,1)$?

Recently I came across the following question: can $H^*(\mathbb{C}P^n;\mathbb{Z})$ be the integral cohomology ring of some Eilenberg-Maclane space $K(\pi,1)$? I guess (without strong evidences) that ...
5
votes
1answer
186 views

Representing classes in *relative* homology by submanifolds

There are nice results for representing homology classes by submanifolds, in particular for any class in $H_i(X)$ with $i\le 6$, see here. When $X$ is low-dimensional I can start getting explicit, but ...
5
votes
0answers
120 views

Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is \begin{equation} \Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in ...
3
votes
0answers
90 views

Maslov class as a relative cohomology class in H^2(M, L)

Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to ...
6
votes
5answers
2k views

(infinity,1)-categories directly from model categories

Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" ...
11
votes
1answer
342 views

Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf. Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
1
vote
1answer
144 views

Orientability of Surfaces and the Fundamental Group [closed]

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
3
votes
1answer
125 views

Connection between Stalling's end theorem and Seifert-van Kampen Theorem

Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
4
votes
1answer
250 views

Homotopy theory of acyclic categories

Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ...
22
votes
1answer
1k views

Example of 4-manifold with $\pi_1=\mathbb Q$

This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
16
votes
1answer
2k views

Example of fiber bundle that is not a fibration

It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I ...