All Questions
9,056 questions
8
votes
2
answers
723
views
Could there be any homotopy group without "Lebesgue Number Lemma"?
This is about a comment that I have made in my general topology class while I was proving the abovementioned lemma as a consequence of compactness!
As far as I know, essentially, there is only one ...
- 20k
3
votes
1
answer
402
views
The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction
$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$
descends ...
- 448
5
votes
1
answer
179
views
Euler class in center of mod 2 Morava K-theory?
I consider Morava K-theory at the prime $p=2$ and height $n$. $K(n)^*$ is multiplicative and complex-oriented, but the multiplication is not commutative. Suppose I have a complex bundle $E$ of rank m ...
- 56.9k
6
votes
1
answer
186
views
Plus construction of the product spaces
I am newly learning plus construction in topology. My question is how to prove the following:
The plus construction of the product of two CW complexes is homotopically equivalent to the product of ...
- 11.2k
5
votes
2
answers
442
views
Sheaf of chain complexs glued by chain homotopy equivalences
Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose:
For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
- 1,611
3
votes
1
answer
351
views
How to define relative orientation in terms of (co)homology?
Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon ...
- 3,948
3
votes
1
answer
135
views
Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
CommunityBot
- 1
0
votes
0
answers
92
views
About filtration of the Leray-Serre spectral sequence
In the following proof, it is used the spectral sequence of the Borel
fibration $X\longrightarrow X_{T}\longrightarrow B_{T}$.
I don't understand how the map $\psi $ is obtained, and how is it an $R$-...
- 1,367
4
votes
1
answer
514
views
A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,367
6
votes
0
answers
150
views
Conceptual proof of Jacobi-like identity for Toda brackets
In the paper $p$-primary components of homotopy groups IV, Toda proved an identity for his bracket operation, which can be succinctly written as
$$[[\alpha, \beta, \gamma], \Sigma \delta, \Sigma \...
- 1,263
2
votes
0
answers
81
views
A question about the Leray-Serre spectral sequence of the Borel fibration
Let $G$ be a torus which acts on a topological space $X$. Then consider
the Borel fibration $X\longrightarrow X_{G}\longrightarrow B_{G}$. Let $%
\left( E_{r}^{\ast ,\ast },d_{r}\right) $ be the Leray-...
- 1,367
4
votes
1
answer
297
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 8,803
4
votes
1
answer
225
views
Homotopy of Brown-Gitler spectra
Let $A^\vee = \mathbb{F}_2[\bar\xi_1, \bar\xi_2, ...]$ be the mod-2 dual Steenrod algebra. One can define a weight filtration on $A^\vee$ by setting $wt(\bar\xi_i)=2^i$ and $wt(xy)=wt(x)wt(y)$. There ...
- 56.9k
7
votes
1
answer
202
views
Lipschitz bounds and homotopy groups of diffeomorphism groups
Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
- 44.4k
2
votes
0
answers
157
views
Symmetric powers for a short exact sequence of vector bundles
If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the ...
- 2,964
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
- 371
1
vote
0
answers
111
views
Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
- 11
1
vote
0
answers
125
views
Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$
Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
- 63.9k
14
votes
1
answer
813
views
What properties of the fundamental group functor are needed to uniquely determine it upto natural isomorphism?
Consider a functor from pointed topological spaces to groups, which
evaluates the same on homotopically equivalent topological spaces and also on homotopic continuous functions.
What additional ...
- 15.9k
7
votes
2
answers
2k
views
Is there a theorem showing that de Rham homology is isomorphic to singular homology?
The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology.
Now de ...
- 12.9k
1
vote
0
answers
46
views
Optimal transport and the geometry of singular measures on fractal Sets
Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
- 11
1
vote
0
answers
37
views
Asymptotic growth of twisted alexander polynomials and hyperbolic volume for infinite families of knots
Let $\{K_n\}_{n=1}^\infty$ be an infinite family of hyperbolic knots with increasing crossing number, and let $\rho_n: \pi_1(S^3 \setminus K_n) \to SL_N(\mathbb{C})$ be a sequence of irreducible ...
6
votes
3
answers
395
views
Decomposable maps of half-smash products
[Cross-posted from MSE]
For a pointed space $X$ and unpointed space $Y$, recall the half-smash product $X\rtimes Y=X\land Y_+=(X\times Y)/(\ast\times Y)$. For unpointed spaces $X,Y$ and a pointed ...
- 121
6
votes
1
answer
313
views
Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?
Let $p$ be a prime and $n > 1$ a height. My conventions for Morava K-theory are that $K_p(n)^*(pt)=\mathbb{F}_p[v_n,v_n^{-1}]$, $|v_n|$ (the degree of $v_n$) is $2(p^n-1)$.
Question: If $G$ be a ...
- 18.2k
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 4,713
5
votes
0
answers
175
views
Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
6
votes
0
answers
155
views
How to characterize this condition for commutative squares in $\Delta$
In the simplex category $\Delta$ we have the situation, that
pullbacks exist for cospans $[a] \xrightarrow{\alpha} [n] \xleftarrow{\beta} [b]$ in $\Delta_\text{mono}$ and
pushouts exist spans $[a] \...
4
votes
1
answer
193
views
Canonical decomposition as wedge sum up to homotopy equivalence
I am curious: is there a canonical way to decompose a finite simplicial complex into a wedge sum up to homotopy equivalence? More formally:
Let $X$ be a finite simplicial complex. Is $X$ homotopy ...
- 16.2k
6
votes
1
answer
360
views
On connected sum of compact manifolds along a submanifold
Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
- 2,479
106
votes
4
answers
13k
views
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
- 12.9k
14
votes
2
answers
1k
views
Derived topological stacks?
I apologize for the vagueness of the following.
Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
- 1,088
14
votes
1
answer
573
views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
- 6,215
17
votes
1
answer
3k
views
The homology of the orbit space
Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper).
Is there a way to understand the homology ...
CommunityBot
- 1
8
votes
0
answers
287
views
What is the current research situation of the Cheeger–Goresky–MacPherson conjecture?
In [CGM-1983], J. Cheeger, M. Goresky and R. MacPherson conjectured that the intersection cohomology of a singular complex projective algebraic variety $X$ is naturally isomorphic to its $L^2$-
...
- 12.9k
4
votes
1
answer
293
views
Relationship between infinite suspension $\Sigma^{\infty}$ of $E_{\infty}$ grouplike space and its infinite delooping?
For an object $X$ in the infinity category of pointed space $S_{*}$, if it has an $E_{\infty}$ grouplike structure, then it give rises to a unique infinite delooping $BX$, which is a connective ...
- 1,467
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 3,948
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 35.5k
12
votes
0
answers
482
views
What is the infinity category of subspaces of $\mathbb{R}^n$?
Let $\mathcal{J}$ denote the topological category of finite-dimensional real inner product spaces with linear isometric embeddings. The space of morphisms $\mathcal{J}(\mathbb{R}^k, \mathbb{R}^n)$ is ...
- 901
3
votes
0
answers
110
views
String cobracket and co-Hochschild homology
Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of ...
- 985
2
votes
0
answers
146
views
Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 356
11
votes
2
answers
594
views
In Top, *how* do conjugate homorphisms of groups induce homotopies of classifying maps?
In Top, how do conjugate homorphisms of groups induce homotopies of classifying spaces?
They exist— there's an abstract proof, but how is BG → BH written in terms of the homotopy??
If G and H are only ...
- 12.9k
4
votes
1
answer
496
views
Homotopy groups of the space of diffeomorphisms
Let $M$ be a smooth, closed, and connected manifold of dimension $n \geq 5$. Let $\operatorname{Diff}(M)$ denote the space of diffeomorphisms of $M$ with the $C^\infty$-topology.
Is there a general ...
- 356
5
votes
1
answer
221
views
Serre spectral sequence of Borel construction
Let $G$ be finite $p$-group, $X$ be path-connected $G$-space, $E=EG\times_{G}X$ be the Borel construction and $BG$ be the classifying space of $G$. Consider Serre spectral sequence of the fibration
$$...
- 44.8k
6
votes
0
answers
136
views
Second homotopy group of the symmetric power of a space
Let $X$ be a finite CW complex, $n \ge 2$, and $\Sigma_n$ be the permutation group on $n$ symbols. Let $X^{(n)}=X^n/\Sigma_n$ be the quotient of the natural action of $\Sigma_n$ on $X^n$. We call $X^{(...
- 311
21
votes
2
answers
2k
views
A simplicial complex which is not collapsible, but whose barycentric subdivision is
Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the ...
CommunityBot
- 1
4
votes
0
answers
148
views
Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
CommunityBot
- 1
2
votes
1
answer
664
views
How to define 0-sphere in a category with zero object?
The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \sqcup \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, on $...
- 48.8k
3
votes
0
answers
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 63.9k
8
votes
1
answer
470
views
Non-triviality of Whitehead products in wedges of CW-complexes
Suppose $X$ and $Y$ are finite, simply connected, based CW-complexes and $m,n\geq 2$. If $a\in \pi_m(X)$ and $b\in \pi_n(Y)$, then one can regard these as elements of the homotopy groups of $X\vee Y$. ...
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