All Questions
9,056 questions
13
votes
1
answer
980
views
Hopf algebras as cohomology of $\mathbb{CP}^\infty$, $\Omega S^3$ and related $H$-spaces
Let me begin by a couple of questions :
Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra ...
- 52.6k
5
votes
1
answer
799
views
Exotic spheres detected in higher homotopy
Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space \...
- 2,734
5
votes
2
answers
1k
views
Generator of a Fukaya category with certain properties
There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a ...
- 21k
20
votes
2
answers
1k
views
How many model categories have the same weak equivalences?
There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model ...
- 4,588
5
votes
1
answer
466
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nerves of crossed complexes, group T-complexes and classifying spaces
A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.
There are a couple of ...
CommunityBot
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7
votes
3
answers
966
views
If a colimit of distinguished triangles exists, is it also a distinguished triangle?
Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps ...
- 1,407
4
votes
3
answers
348
views
Cohomological dimension of a group acting on a cellular complex
Let $G$ be a group acting on $X$, $X$ a cellular complex and $cd(G)$ the cohomological dimension of $G$.
2 things:
(1) I'm looking for a reference (or proof!) of this:
Suppose $X$ is acyclic. Then $...
- 35.9k
2
votes
1
answer
350
views
Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces
Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi_n(Y) \to \pi_{n-1}(\...
- 12.5k
8
votes
1
answer
1k
views
Topological degree of polynomial maps.
The $\mathbb{Z}_2$ topological degree of a (non-constant) polynomial in one variable, clearly, coincides with its degree as a polynomial, $\mod 2$.
Consider further a polynomial self-mapping $F$ on ...
- 4,586
5
votes
1
answer
455
views
What immersed closed curves on the double-torus are non-trivial when lifted to the unit tangent bundle?
Take an equator on the two sphere $S^2$ and parametrize it by arc-length obtaining a closed loop $\alpha: S^1 \to S^2$. The curve $(\alpha,\alpha'):S^1 \to T^1S^2$ in the unit tangent bundle of $S^2$ ...
- 32.4k
5
votes
3
answers
1k
views
Obstruction Cocycles
Hey everyone, I was reading about obstruction theory, here's a bit of a summary. Take a cellular space $X$ and a fibre bundle $p:E \to X$ with fiber $F$; consider the problem of extending a section $s$...
CommunityBot
- 1
7
votes
0
answers
297
views
Inner product on Hochschild homology in 2d TCFTs
This should be an easy question for some people. Take a compact $A(\infty)$ algebra with a cyclically symmetric non-degenerate inner product. In Kontsevich and Soibelman's article "Notes on $A(\infty)$...
- 3,758
2
votes
2
answers
617
views
Comparing lower central series and augmentation ideal completions
Let G be a group. Let $G^p$ be the completion of G with respect to the mod p lower central series of G.i.e. $G^p=\varprojlim_{q} G/\gamma_qG$, where $\gamma_qG$ is generated by all $\{[x_1,\cdots,x_s]^...
- 22.6k
11
votes
5
answers
3k
views
Ribbon graph decomposition of the moduli space of curves
What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
- 2,734
9
votes
3
answers
1k
views
Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
CommunityBot
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6
votes
3
answers
1k
views
Why are spectra indexed over the natural numbers?
A spectrum is a sequence $X_0,X_1,...$ of spaces together with structure morphisms $\Sigma X_n\to X_{n+1}$. To get the usual model for the stable homotopy category based on the category of spectra, ...
- 2,782
13
votes
3
answers
690
views
Long line fundamental groupoid
This question got me thinking about what makes the fundamental group (or groupoid) tick. What is so special about the circle? As another possible candidate for generalization, what about taking the ...
CommunityBot
- 1
30
votes
3
answers
3k
views
Why is homology not (co)representable?
This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?
- 6,308
1
vote
1
answer
414
views
Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
CommunityBot
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3
votes
0
answers
963
views
How to prove that a map is a Serre fibration?
I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...
- 14.4k
2
votes
0
answers
243
views
topology of infinite union of hyperplanes
Hi all:
I am working on Functional Analysis. I encounter a topology problem in my study of spectrum of certain operators, and it has bothered me for quite some time. Any idea or references is greatly ...
- 60.6k
10
votes
1
answer
1k
views
Leray-Hirsch principle for étale cohomology
Let $p:E\to B$ be a continuous map of topological spaces and set $F_x=p^{-1}(x)$ for an $x\in B$. Take a commutative ring $A$ and assume for simplicity that each $H^\*(F_x,A)$ is a free $A$-module. ...
- 22.6k
15
votes
2
answers
927
views
$\pi_4$ of simply-connected 4-manifold
In Baues "The homotopy category of simply conected 4-manifolds" there is some
algebraic description of $\pi_4(M^4)$ where $M^4$ is simply-connected closed 4-manifold, but this description is pure ...
- 68.9k
7
votes
1
answer
563
views
Reference for base change of cohomology pull-push for clean intersections.
Let $X$ be a compact oriented manifold, and $A$ and $B$ closed oriented submanifolds intersecting cleanly. Then I've always been under the impression that pushing forward a cohomology class from $A$ ...
- 35.9k
8
votes
2
answers
2k
views
Intutive interpretation about Linking forms
Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with
$H_{*}(M;Q)=H_{*}(S^3;Q)$
As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's ...
- 3,901
0
votes
1
answer
423
views
What Is This Quotient Space?
Let $X$ be a finite CW-complex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. ...
- 52.6k
16
votes
1
answer
2k
views
Group Completions and Infinite-Loop Spaces
Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that
$\pi_0(Y)$ is a group
The Pontrjagin ring $H(Y; R)$ is the localization of the ...
- 22.6k
3
votes
1
answer
408
views
Morava's "Motives and cell bundles"?
Hello, do you know more about, or some exposition of Morava's talk?
- 1,554
20
votes
6
answers
2k
views
A canonical and categorical construction for geometric realization
There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a ...
CommunityBot
- 1
11
votes
1
answer
565
views
The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex.
Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all compact subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the ...
- 4,990
12
votes
1
answer
2k
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Conventions for definitions of the cap product
In singular (co)homology, if $\alpha\in C^*(X)$ and $x\in C_*(X)$, then the cap product $\alpha \cap x$ is generally defined by the following process:
Apply to $x$ the diagonal map $C_*(X)\to C_*(X\...
- 55.9k
4
votes
3
answers
2k
views
Connected components of space of maps between two manifolds
Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$?
Specifically, I'm thinking of the ...
- 8,803
4
votes
1
answer
320
views
a naive question about homogeneous polynomials
Suppose $p$ is a homogeneous polynomial in $n$ complex variables. Let S be the hypersurface
defined by $p(z)=0$. Then is the 1-form $dp/p$ always non-exact on the complement $C^n\setminus S$?
Any ...
- 831
0
votes
1
answer
251
views
altering curvature on a tessellation representation of a compact surface
I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...
- 28.2k
9
votes
2
answers
778
views
Is the cohomology of a topological operad a cooperad?
For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in ...
- 44.4k
2
votes
1
answer
307
views
Name for a kind of topological property?
What should I call a property (P) of (open) subspaces of a space $X$ such that:
If $U$ satisfies (P), then so does every open subset $V\subset U$
If {$U_i$} is a pairwise disjoint collection of ...
- 60.6k
13
votes
4
answers
5k
views
Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
- 11
5
votes
1
answer
390
views
Values of the multiplicative group over a ring spectrum
In his notes on elliptic cohomology, Lurie defines the multiplicative group $\mathbb{G}_m$ over a ring spectrum $A$ as $\operatorname{Spec} A[\mathbb{Z}]$. What is the value $\mathbb{G}_m(B)$ of the ...
- 3,726
5
votes
1
answer
572
views
Maurer-Cartan and representable functors on differential graded commutative algebras
Let $\mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $\mathfrak{g}$ defines a ...
- 19.8k
9
votes
1
answer
1k
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How does the Lefschetz-Poincare dual torsion linking pairing on manifolds with boundary interact with the maps of the long exact sequence of the manifold-boundary pair?
I'm wondering if anyone can point me to a reference on how the various
Lefschetz-Poincare dual torsion pairings of a manifold with boundary fit
together.
To explain in more detail, consider a ...
- 5,534
6
votes
1
answer
1k
views
A chain homotopy that does not arise from a homotopy of spaces?
Algebraic topologists like to cook up algebraic invariants on topological spaces in order to answer questions, so they are often concerned with how strong those invariants are. Currently, I am ...
- 1,056
19
votes
5
answers
3k
views
The definition of homotopy in algebraic topology
In this post, let $I=[0,1]$.
Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the ...
- 8,803
4
votes
2
answers
764
views
Action of the mapping class group on middle-dimensional cohomology
Given an even dimensional manifold, the mapping class group acts on middle dimensional cohomology (or homology) and this action preserves the intersection form. For manifold of dimension $4k+2$, the ...
- 13.2k
61
votes
2
answers
3k
views
The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many ...
- 2,181
7
votes
1
answer
710
views
Can string topology be a open-closed TCFT with the full set of branes?
String topology studies the algebraic structure of the homology of the free loop space $LM = Map(S^1,M)$ of a oriented closed manifold. One aspect of this structure is that the pair $(H_\ast(LM;\...
CommunityBot
- 1
4
votes
1
answer
742
views
Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 20.7k
5
votes
0
answers
203
views
Homotopy group of space of gauge fields modulo gauge equivalence on T^4
Singer observed in 1978 (Comm.Math.Phys. 60, 7-12) that the homotopy group of the space of gauge fields modulo gauge equivalence with gauge group $G$ on $S^4$ is given by
$\pi_n({\cal A}/{\cal G}) = \...
- 362
3
votes
2
answers
960
views
X not simply connected and X-x contractible
Hello,
I was wondering if there is a nice counterexample to the following question.
Suppose $X$ is a CW-complex which is not simply connected and there is a point $x\in X$ such that $X-x$ is ...
- 5,137
15
votes
1
answer
2k
views
Are the path components of a loop space homotopy equivalent?
If $X$ is a based space, then we have $\pi_1(X) \cong \pi_0(\Omega X)$. This is to say we can identify elements in the fundamental group of $X$ with path components of the first loop space of $X$. ...
- 20.8k
2
votes
1
answer
1k
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Question related to the moduli space of Riemann surfaces and a fibration
If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1_{g} \...
- 1,499