All Questions
9,056 questions
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150
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Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
1
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0
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194
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Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
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138
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$
Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
- 10.5k
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102
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DA structure of a Dehn twist
I am trying to find the DA bordered homology structure of a Dehn twist.In https://arxiv.org/pdf/0810.0687v6.pdf page 255 bottom the authors tabulate the differentials of the right module(A side) of ...
- 99
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191
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Questions related to Morse theory
I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood ...
- 1,177
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222
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Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
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86
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Terminology: are there any names for "quotients" of cellular towers in stable categories?
A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; ...
- 16.9k
1
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180
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Homotopy section but no pointed homotopy section
Can anyone give an example of a pointed map $p:(E,e)\to (B,b)$ between connected pointed spaces (reasonably nice, say of the homotopy type of CW complexes) such that $p$ admits a homotopy section but ...
- 35.9k
1
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218
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How to check a fiber bundle is trivial
Given a smooth fiber bundle $X \to S^1,$ such that the fiber, $F$, is homotopic to $S^2 \vee S^2.$ Is it true that this is always a trivial fiber bundle?
In general, how to check a fiber bundle is ...
- 1,177
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97
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Homotopy type of complement to a union of linear subspaces
Im not sure if this question is appropriate for MO, but I'm looking for a hint about some questions about homotopy type of complement to a union of linear subspaces in vector space $\mathbb{R}^n, \...
- 11
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90
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Regular mapping space vs continuous mapping space for affine schemes
Let $A$ and $B$ be Zariski open affine sub-schemes of $\mathbb{A}_{\mathbb{C}}^{n}$ and $\mathbb{A}_{\mathbb{C}}^m$ respectively. We denote the infinite symmetric product of $B$ by $Sym^{\infty}(B)$. ...
- 5,901
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146
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Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?
In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...
- 11.8k
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160
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Contractible four-manifold which admits a decomposition
Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is ...
- 891
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56
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Effect on finite transformation semigroup under a particular modification of the generators
The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
- 695
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412
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History of simplicial complex
It is easy to find the definition of a simplicial complex:
https://en.wikipedia.org/wiki/Simplicial_complex
I am interested in the history and first occurrences of the concept.
When did people start ...
- 479
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66
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First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?
In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
- 1,093
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170
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What is this algebraic object (special case of a semigroup)?
Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
- 696
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129
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Open cone homeomorphic to the Euclidean space
Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
- 2,535
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139
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Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
- 12.5k
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121
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Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
- 59
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139
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Free monoids on posets
I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying
if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
- 12.5k
1
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259
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Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
- 51
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0
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99
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topological dimension of inverse limit of compact spaces
Let $X$ be a compact metric space with topological dimension ${\rm dim}(X)>0$. Let $f: X\times X\to X$ be a continuous and surjective map. Define a family of maps $f_n: X^{n+1}\to X^{n}$ for $n\ge ...
- 675
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0
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298
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Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
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271
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Eilenberg-Steenrod cohomological theory versus Weil cohomological theory [closed]
Can someone enlighten me what is the difference between an Eilenberg-Steenrod cohomological theory ( See here, https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms ), and a Weil ...
- 595
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231
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transition in homotopy theory
I guess that the following are true; maybe classical? Is there a reference?
Let $X, Y, Z$ be connected pointed CW-complexes, $f:X\to Y$ and $g:X\to Z$ pointed maps. Assume that
for every $k\ge 1$, ...
- 2,479
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0
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213
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Defining path on the prime spectrum
If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...
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0
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151
views
Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
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101
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Rational systole of a manifold
I also posted this question on MSE, but since it may be a delicate question, I decided to post it here.
Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ ...
- 2,095
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0
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80
views
A characterization for a space that is similar to locally connected spaces
Let $X$ be a $T_0$ topological space with the property that there exists a basis $\{O_i\}_{i\in I}$ for $X$ such that for each $J\subseteq I$ the subspace $\bigcap_{i\in J}O_i$ has only finitely many ...
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0
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85
views
Vanishing cycles and relative homology of I_n fiber
Suppose $\pi: X \rightarrow D$ is a smooth elliptic fibration with a section over the closed disk without multiple fibers, such that all special fibers are in the interior of $D$. The boundary $\...
- 67
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0
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83
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What is known about the algebraic completion of a monoid?
It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid:
Let $W$ be a monoid and let $p(x)=q(...
1
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0
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231
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Has an "algebraic manifold" been defined before? Are there any non-trivial examples?
Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds:
for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $...
1
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0
answers
102
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What is the real name for the initial object in the category of "monoid-valued measures of intervals" on transitive relations?
(I'm not asking for a true/false answer; I have a true algebraic fact and I'm looking for a reference in the literature. By the way, there is a version of this theorem that replaces monoid with $R$-...
1
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0
answers
52
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Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?
The terms are defined in a related question. [1]
Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
1
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0
answers
71
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terminology for a kind of two-sided module over a monoid
If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are ...
- 12.5k
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179
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Zeroth cohomology of tensor product of complexes concentrated in nonpositive degrees
This is probably an easy problem, but I can't find any reference.
Let $V$ and $W$ be cochain complexes over some commutative ring, and assume that they have both cohomologies concentrated in ...
- 2,079
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0
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295
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How to calculate the total chern classes of CP^n [closed]
When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$:
$$\ 0\to S \to C^{n+1} \to Q \to 0$$
And identify the bundle $\ TCP^...
- 157
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0
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331
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why is monodromy weight filtration compatible with cup product?
This question is about a statement I took for granted in this question.
If $f: X \to S$ is a moprhism from a complex manifold to a punctured disc then the monodromy operator $T$ is quasi-unipotent, so ...
- 4,070
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0
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231
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Cohomology of realization space of matroid
Do we know any thing about cohomology of realization space of matroid (the space of all set of vectors in $\mathbb{C}^k$ which captures the independence structure of matroid $M$), more simple, for ...
- 91
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0
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244
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A $d_1$-differential in the homotopy fixed points spectral sequence
I have the following problem. Let $Q$ denote $\mathbb{Z}/2$ as an abelian group and let $X$ be a $Q$-spectrum. If I want to compute the homotopy groups of $X^{hQ}$, homotopy fixed points spectrum, I ...
- 1,759
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355
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On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
- 494
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0
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133
views
Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
- 5,901
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0
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163
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Explicit map between $\check{H}^1(M,\underline{\mathbb{R}})$ and $H^1(M,\mathbb{R})$
Is there a way to construct an explicit isomorphism between Cech cohomology and singular cohomology on a smooth manifold for degree 1? If yes can this be extended to higher degee?
- 789
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152
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Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
- 11
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0
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154
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Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
- 19
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0
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137
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Covers of a 4-manifold pull back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
Is ...
user164740
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0
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213
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Is an (n-1)-sphere quotient by an (n-1)-sphere contractible? [closed]
I am thinking about the homotopy type of the following quotient space:
Let $X$ be a topological space and $A$ be a subspace of $X$. If both $X$ and $A$ have homotopy type of a sphere $S^{n-1}$ (of the ...
- 461
1
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0
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129
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Nontrivial integer homology class implies orientability
I posted this question on MSE and I would like to see if my reasoning is correct.
Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and ...
- 2,095
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0
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107
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What can be said about the Chow rings of classifying spaces of semi-direct products of groups?
For instance, what can we say about the Chow ring of the classifying space of a semi-direct product $CH^*(B(G\ltimes H))$, in terms of the Chow rings of $CH^*(BG)$, $CH^*(BH)$, and the singular ...
- 935