Homotopy, stable homotopy, homology and cohomology, homotopical algebra.
11
votes
0answers
141 views
Are there geometrically formal manifolds, which are not rationally elliptic?
Formality of a space is meant in the sense of Sullivan, e.g. 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
14 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 ...
10
votes
2answers
371 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 ...
9
votes
0answers
106 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 ...
10
votes
0answers
253 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 ...
10
votes
4answers
459 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
62 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
124 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
198 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
0answers
88 views
Fundamental group of row of spheres [migrated]
The fundamental group of $S^1$ is $\mathbb Z$. Let's also call that space $P_1$. Then we'll build $P_n$ for $n > 1$ by taking $P_{n-1}$ and adjoining a circle to it with the condition that it must ...
1
vote
2answers
207 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
vote
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
141 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
$$e^{i\theta} \cdot (z_0,z_1,z_2) = ...
7
votes
1answer
191 views
Homotopy of localisations of colimits
Let $X_k$ be a family of spectra equipped with maps $f_k: X_k \to X_{k+1}$. If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy ...
2
votes
2answers
150 views
Canonical n plane bundle over Lagrangian Grassmanian
Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...
9
votes
0answers
211 views
Topological type of Brieskorn manifolds
Let us consider the complex hypersurface and suppose that $n\geq 3$:
$$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
393 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
145 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
159 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
93 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
261 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
votes
0answers
187 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 ...
9
votes
2answers
346 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
120 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 ...
3
votes
1answer
388 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
166 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
254 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
95 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
vote
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
194 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
203 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 ...
9
votes
1answer
255 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.
...
4
votes
1answer
135 views
Semidirect product of torus with cyclic group: representations/cohomology?
Let $p$ be prime and let $T^p$ be the $p$-torus and $\mathbb{Z}/p$ the cyclic group of order $p$ generated by $(12\ldots p)$. Consider the semidirect product $T^p\rtimes\mathbb{Z}/p$ with the natural ...
3
votes
1answer
89 views
Relative flasqueness?
It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
7
votes
1answer
269 views
Two questions about sphere bundles
I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and ...
2
votes
1answer
352 views
A question about the universal coefficients theorems
This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...
11
votes
1answer
283 views
Are totally degenerate chains null-homologous?
Let $X$ be a CW complex.
Suppose $\gamma\in C_n(X)$ is a cycle which is a sum of maps $\sigma:\Delta^n\to X$ which factor as $\Delta^n\to\mathbb R^{n-1}\to X$. Does it follow that $\gamma$ is ...
20
votes
5answers
1k views
(Short) Exact sequences with no commutative diagram between them
This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...
10
votes
0answers
316 views
Singular chains generated by manifolds with corners — does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
2
votes
1answer
116 views
Seifert fiberable manifolds with several Seifert fiberings
I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds:
Hatcher: Notes on Basic 3-Manifold Topology.
Regarding the class d), it follows from Proposition 2.1 on page 31, that ...
0
votes
0answers
59 views
Problem on infinite dimensional metric space, with rigidity assumption
By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
...
1
vote
0answers
73 views
PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
19
votes
0answers
464 views
On a homological finiteness condition
Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated.
Question: does there exist a finite CW complex $Y$ and a map ...
1
vote
0answers
46 views
Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
7
votes
2answers
255 views
Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex
Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...
9
votes
0answers
213 views
Cohomology and impossible figures
In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": ...
1
vote
2answers
232 views
Equivalence of homotopy categories and model structure theory
I apologize if this is to basic for MO, but it just seems a bit to advanced for SE.
Let $\text{Top}$ be the category of topological spaces and $\text{sSet}$ the category of simplicial sets. There is ...
8
votes
0answers
156 views
Homology of compact symmetric spaces
Could anyone point me to a table giving the homology of all compact symmetric spaces, i.e. $G/U$ where $G$ is a compact Lie group and $U$ the fixed points of an involution of $G$? I'd be happy even ...
2
votes
0answers
147 views
Homotopy groups of the classifying spaces BO and BTOP
I was reading a proof and it concludes, from the fact that the rational Pontryagin classes are topological invariants, that the map $\pi_{4n}(BO)\rightarrow\pi_{4n}(BTOP)$ induced by the inclusion ...
9
votes
1answer
279 views
Are these two notions of “dualizable” spectra equivalent?
A spectrum $X$ is dualizable if the natural map $$Map(X,\mathbb S) \wedge X \rightarrow Map(X,X)$$ is an equivalence of spectra. This is equivalent to having evaluation and coevaluation maps in the ...
