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

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13
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2answers
748 views

Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
1
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1answer
174 views

Pushout of categories along embeddings gives homotopy pushout?

Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve ...
10
votes
0answers
374 views

Has this chain complex associated with a simplicial complex been studied before?

I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference. Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...
-1
votes
0answers
40 views

Euler characteristics and 2-manifold mesh [migrated]

I would like to ask you for a simple definition of two-manifold surfaces and correlation with Euler characteristics $\chi=2-2g$. Is this characteristics always valid for 2-manifolds? More particularly ...
15
votes
1answer
387 views

Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
1
vote
2answers
221 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
2
votes
0answers
122 views

Canonical trivialization of Stiefel-Whitney classes on the frame bundle

Let $X$ be a smooth manifold, perhaps oriented if necessary. The frame bundle $\pi:P \to X$ carries a canonical trivialization of the pullback of the tangent bundle of $X$ and thus a canonical ...
8
votes
1answer
298 views

Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
1
vote
1answer
218 views

Singular cohomology as a Zariski sheaf

Let $X$ be a complex algebraic variety, and consider the presheaf $U \mapsto H^i(U^{an}, \mathbb Z)$ in the Zariski topology. Is there a theorem that says this presheaf is already a sheaf, for ...
9
votes
1answer
348 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
-3
votes
1answer
254 views

Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given. We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$. For ...
0
votes
0answers
17 views

Topology on the set of infinite sequences [migrated]

Help, who can give me an example to show the following definition of $lim$ and $closed \ set$ let $\Sigma$ be a set of states, let $\Sigma^\omega$ denote the set of all infinite sequences of elements ...
1
vote
1answer
163 views

How to classify continuous/differentiable maps from $T^2$ to $U(N)$?

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...
2
votes
0answers
103 views

Cylinder isotopy relative to its bases

Is n-times twisted cylinder isotopic to 0-times twisted cylinder relative to its bases? (I hope I got the terms right) That is, is it possible to twist (stretching, bending etc. is allowed) a cylinder ...
17
votes
1answer
365 views

What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$?

What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$? The group structure induced by the group structure on the codomain. This question is a followup to Eric's answer ...
7
votes
1answer
187 views

worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...
9
votes
1answer
277 views

Can the group of homotopy classes of maps into S^3 be noncommutative?

Let $[X,S^3]$ be the set of homotopy classes of maps from the pointed CW complex X to the pointed 3-sphere. The group structure on S^3 endows this set with a group structure. Is this group ever ...
1
vote
1answer
306 views

A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.) Theorem (Quillen) ...
14
votes
1answer
353 views

Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...
0
votes
0answers
27 views

Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded?

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...
11
votes
2answers
415 views

Postnikov's algebraic reconstruction of cohomology from homotopy invariants

In his short paper (1951) and longer monograph (1955), Postnikov introduced what I believe are now called Postnikov systems or towers. It is my understanding that Postnikov systems have since then ...
10
votes
0answers
150 views

A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
12
votes
0answers
301 views

Classical and Quantum Chern-Simons Theory

Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway. Let $\Sigma$ be a two-manifold and $M$ a ...
12
votes
4answers
599 views

Functors and coverings

A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...
3
votes
1answer
68 views

Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...
4
votes
1answer
139 views

The Borel construction of equivariant cobordism

In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ...
7
votes
2answers
211 views

Cokernel of the stable J-homomorphism at odd primes

Where can one learn about odd-primary components of the cokernel of the stable J-homomorphism? According to wonderful Wikipedia article on Homotopy groups of spheres, the "hard" part of the stable ...
1
vote
2answers
221 views

A generalized Thom Isomorphism

The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section ...
1
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0answers
80 views

Euler class and self-intersection number of a surface in a 4-manifold [duplicate]

In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that For a compact oriented surface $X$ in a 4-dimensional oriented ...
2
votes
1answer
154 views

Chern class of Hopf fibration over elliptic curve

Let $N = \{ (z_0,z_1,z_2) \in S^5 \mid z_0^3+z_1^3+z_2^3 = 0 \}$, where we consider $S^5\subset\mathbb{C}^3$. The circle $U(1)$ acts on $N$ by $$e^{i\theta} \cdot (z_0,z_1,z_2) = ...
7
votes
1answer
199 views

Homotopy of localisations of colimits

Let $X_k$ be a family of spectra equipped with maps $f_k: X_k \to X_{k+1}$. If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy ...
2
votes
2answers
160 views

Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...
9
votes
0answers
230 views

Topological type of Brieskorn manifolds

Let us consider the complex hypersurface and suppose that $n\geq 3$: $$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$ and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...
2
votes
1answer
408 views

Is there a direct proof of the following real analysis fact?

I want to prove the following fact without using topological degree theory or related algebraic topology Let $h:\overline{B}(0,1)\to \mathbb{R}^n$ be a continuous map such that $|h(x)-x|\leq \delta$ ...
1
vote
1answer
161 views

Is a 'join' of two cofibrations a cofibration?

I have encountered with following problem while I was learning homotopy theory. Let $A\rightarrow X$ and $B\rightarrow Y$ are cofibrations (in a good category). Is it true that a map $A\times ...
2
votes
1answer
171 views

fundamental group and torus action

Let $T$ be the complex torus acting on a complex connected algebraic variety $X$ and let $p \colon X\rightarrow Y$ be a good quotient for this action. For any $y\in Y$ we have a sequence $p^{-1}(y) ...
7
votes
0answers
102 views

Cosimplicial commutative rings in stable homotopical algebra

When doing "derived commutative algebra" over a discrete commutative ring $R$ that doesn't contain $\mathbb{Q}$, it's fairly well known that you generally have two flavors of "totally commutative ...
3
votes
1answer
290 views

Relations between Stiefel-Whitney classes

I need to know all relations between Stiefel-Whitney classes for closed manifolds of dimensions 3 and 4. Unfortunately, I found the literature on the subject quite confusing. The answer for all ...
7
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0answers
208 views

Characteristic Classes in Geometric Representation Theory

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used. I wonder if there are concrete applications of the ...
10
votes
2answers
369 views

Restriction of “$\pi_{1}$” to topological groups

Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?
2
votes
0answers
138 views

What kinds of manifolds admit non-vanishing vector fields defining convergent congruences?

One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$. I am ...
4
votes
1answer
412 views

Topological fundamental group of spec(R)

Let $R$ be a commutative ring with identity. Assume that $X = Spec(R)$ with the Zariski topology. When is this space path connected? And also we want to know the topological fundamental group of ...
5
votes
0answers
178 views

Homology of stack points (from math.stackexchange)

(This question is duplicated here) This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the ...
6
votes
1answer
264 views

Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity. Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...
4
votes
0answers
111 views

Projective representation of diffeomorphism group of $S^2$

We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class. Then do we have a classification of the ...
1
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0answers
31 views

Equivalence between the definitions of Reidemeister Coincidence number

Given two applications $f$ and $g$, denote by $R (f, g)$ the set of Reidemeister classes determined by $f$ and $g$ (according to the algebraic definition, on the induced on fundamental groups). And ...
2
votes
0answers
203 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) ...
0
votes
1answer
214 views

Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
3
votes
4answers
297 views

What does it mean to speak of a homotopy fibration sequence?

I'm reading a paper in which the following is done. We have a certain particular map of spaces $f:X\to Y$ and then it is said something along the lines of "let $Z_f$ denote the space whose defining ...
10
votes
1answer
260 views

Can we use unparameterized chains to calculate singular homology?

Most models of singular chains on a topological space $X$ use maps from some particular collection of "nice" objects, such as the standard simplices $\Delta^n$, the standard cubes $[0,1]^n$, etc. ...