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

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43 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

**0**answers

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 $\mathbb{C}^n$, $n \ge 2$. ...

**3**

votes

**2**answers

93 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 ...

**19**

votes

**2**answers

669 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 ...

**3**

votes

**0**answers

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

**1**answer

102 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

**1**answer

304 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 ...

**2**

votes

**1**answer

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 ...

**35**

votes

**1**answer

791 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 ...

**5**

votes

**2**answers

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 ...

**3**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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 ...

**0**

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**0**answers

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?

**12**

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**1**answer

211 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
$$X_0\rightarrow X_1\rightarrow\ldots$$
$$Y_0\rightarrow Y_1\rightarrow\ldots$$
...

**0**

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**0**answers

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 ...

**0**

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**0**answers

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 ...

**4**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

**1**

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**1**answer

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**0**answers

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 ...

**5**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 ...

**3**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**5**

votes

**1**answer

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 ...

**4**

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**0**answers

127 views

### Adams Spectral sequence and Pontrjagin-Thom construction [Reference request]

I will be grateful for any reference for the following statements/claims.
1) Let's consider the case of $p=2$ and the classic Adams spectral sequence with the $E_2$-term given by ...

**-1**

votes

**0**answers

68 views

### diagonal stratum in symmetric product

Fix a Riemann surface $(\Sigma, j )$ and a partition $\pi: r=\Sigma a_in_i$ for integer $a_i \ge 1, n_i \ge 1$, there is a diagonal stratum $\chi_{\pi}$ indexed by $\pi$ comprising the image of the ...

**3**

votes

**1**answer

192 views

### Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits.
Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...

**13**

votes

**3**answers

656 views

### Simply-connected rational homology spheres

Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in ...

**0**

votes

**1**answer

189 views

### Torsion in the (co-)homology of a smooth projective variety - what is known in general?

There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...

**6**

votes

**1**answer

225 views

### Four-dimensional vector bundles over $S^4$, intuition?

I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...

**6**

votes

**0**answers

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.

**15**

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**1**answer

316 views

### The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...

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**0**answers

187 views

### solid commutative ring spectra

Let $R$ be a discrete (i.e. an ordinary) commutative ring and let $HR\rightarrow T$ be a map of $E_{\infty}$-ring spectra where $HR$ is the associated Eilenberg-Mac Lane ring spectrum. We say that $T$ ...

**3**

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**0**answers

71 views

### What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [migrated]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...

**0**

votes

**1**answer

189 views

### When is the Thom class the Poincare dual of the zero section?

As the title suggests, when is the Thom class the Poincare dual of the zero section? For starters, it's true for the normal bundle of an immersion...

**3**

votes

**1**answer

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 ...

**3**

votes

**1**answer

186 views

### Constructing a homology class of degree $d(d-1)/2$ in $H_3(S^3)$

There is a nice construction of a class of degree $d^2$ in $H_3(S^3)$. Take a class $h$ of degree $d$ in $H_1(S^1)$, and then take its join with itself: $h*h$ is degree $d^2$ in $H_3(S^1*S^1)$, and ...

**2**

votes

**0**answers

116 views

### the homology of configuration spaces [closed]

In the paper ON THE HOMOLOGY OF CONFIGURATION SPACES, Section 5.4,
Why $C(M\times \mathbb{R};X)\simeq \Omega C(M; SX)$? Does this hold for $X$ not connected?

**0**

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**0**answers

66 views

### Text book for sheaf theory [migrated]

Is there any nice text book for sheaf theory for an under gradute student? Tennison's sheaf thory was too hard for me, Please help me, Thanke you very much.

**1**

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**0**answers

81 views

### When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...

**6**

votes

**1**answer

240 views

### Fredholm operators in $K$-theory?

Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...