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

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123 views

Covering space theory, category theory

Requiring covering spaces of a well-behaved connected topological space $X$ to be connected, let $\mathcal{Cov}(X)$ be the category of covering spaces of $X$ and maps over $X$ and maps over $X$. Can ...
10
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134 views

Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
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0answers
80 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$ [migrated]

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
3
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1answer
130 views

characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space $$ B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2 $$ where $$ \Delta=\{(m,m)\mid ...
3
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1answer
221 views

characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal $$ ...
4
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1answer
103 views

Stable cohomology operation, natural homomorphism

How do I see that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?
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99 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ ...
4
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1answer
100 views

cohomology ring of infinite iterated loop space

What is the cohomology ring $$ H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)? $$ I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
4
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1answer
153 views

Physical strength of a link [on hold]

Assuming that we construct a link/chain using a collection of knots. Is there a way to measure the physical strength of this chain?
23
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219 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here. $\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an ...
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66 views

cohomology ring of base-point-preserving maps on the 3-sphere

I find that $\text{Map}_*(S^3;S^3)=\Omega^3S^3$. I want to find the cohomology ring of $H^*(\Omega^3S^3;\mathbb{Z}_2)$. In the paper On configuration spaces, their homology, and Lie groups, I find ...
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2answers
201 views

bar construction and loop space cohomology

Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that ...
4
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1answer
87 views

unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ ...
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87 views

Fundamental Group of SL_2 [on hold]

I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ...
2
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1answer
121 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 ...
4
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1answer
115 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 ...
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0answers
57 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 $\mathbb{C}^n$, $n \ge 2$. ...
4
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2answers
148 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 ...
23
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2answers
908 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 ...
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0answers
41 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 ...
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1answer
118 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 ...
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1answer
330 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 ...
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1answer
971 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 ...
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2answers
230 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 ...
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1answer
178 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 ...
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1answer
109 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 ...
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75 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?
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1answer
265 views
+50

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 $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
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92 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 ...
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125 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 ...
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96 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
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1answer
213 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
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1answer
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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
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1answer
182 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 ...
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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 ...
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130 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. ...
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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
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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 ...
6
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1answer
144 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 ...
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50 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, ...
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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 ...
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0answers
121 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 ...
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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 ...
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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
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1answer
127 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 ...
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1answer
146 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
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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 ...
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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 ...
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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
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1answer
194 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 ...