All Questions
9,056 questions
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211
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reference for homology complex projective space
I am looking for references on homology complex projective spaces; or more precisely the classification (if any) of smooth oriented manifolds which have the same homology groups as $\mathbb{CP}^n$.
anon
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0
answers
379
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Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
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163
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Presentation complex of a finite perfect group and its features
Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...
- 1,177
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54
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On the fundamental dimension of a polyhedron
Fundamental dimension of a finite polyhedron $P$ is defined as : $$Fd(P)=\min \{ \dim (X):X\; \text{and} \; P \; \text{have the same homotopy type}\}.$$
My question is that: if $A$ is homotopically ...
- 1,182
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107
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What is the average degree of a d-simplex?
I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...
- 1
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195
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Definition of union of simplicial complex and a subset
(Cross-posted from MSE: https://math.stackexchange.com/questions/4425225/definition-of-union-of-simplicial-complex-and-a-subset)
Consider a simplicial complex $\Delta$ with vertex set equal to some ...
- 521
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336
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Can someone explain this proof on aspherical manifolds?
I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is:
...
- 29
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57
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Given m vectors in n dimension where m>>>n, how do you find the vectors that define the largest convex hull constructed with the vectors?
Say there are m vectors in n dimensional space (m>>>n).
There exists a largest convex hull defined by a subset of those vectors.
My goal is to describe the space that is strictly inside the ...
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42
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Conditions on a set implying properties on neighborhoods
Suppose $F$ is a closed set in a Euclidean space, and for $\epsilon>0$, let $V_\varepsilon$ be the $\varepsilon-$neighborhood of $F$ i.e. the set of points $x$ having a distance less than $\...
- 411
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260
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Another definition of singular homology
The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows:
Let $X$ be a topological space. A $n$-...
- 781
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266
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Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
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143
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Betti number of the boundary of a 4-manifold
Let $M$ be a compact $4-$manifold with boundary $dM$. If $M$ has the homotopy type of a wedge of $2-$spheres then is it always true that $b_1(dM)=0$? By $b_1$, I mean first Betti number.
It is known ...
- 1,177
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74
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Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 9
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99
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Spaces of $n$-dimensional topological spaces whose fundamental group is given
If we fix a group $G$ and a dimension $n$, we can ask which $n$-dimensional locally path-connected$^*$ (or otherwise sufficiently nice) topological spaces $X$ have $\pi_1(X) \cong G$. Would these ...
- 79
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88
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Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]
Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
- 10.5k
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133
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Covering map preserved under homotopy equivalence
Given a $m-$sheeted covering map from $p:M^n\to N^n$, where $M,N$ are manifolds of dimension $n$. Suppose $M$ and $N$ are homotopy equivalent to finite CW complexes $X$ and $Y$ of same dimension $k$. ...
- 1,177
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171
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A fibration is a map which has the right lifting property with respect to injections that are weak equivalences
As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram
$$\require{AMScd}
\begin{CD}
A @>>> X;\\
@VV{i}V @VVV \\
B @>&...
- 1,168
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179
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Adjunction formula for non compact surfaces
Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
- 25
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170
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Cohomology ring of 5-manifold generated in degrees 1, 2, 3
Is there a connected closed 5-manifold $M$ such that $\oplus_{i\geq 0} H^i(M, \mathbb{Z})$ is generated by $H^1(M, \mathbb{Z})\oplus H^2(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$ but is not generated ...
- 21
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293
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Quotient of monoids and monoid algebras
Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
- 327
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105
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Flag variety as monoid and Schubert calculus
The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...
- 11
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250
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Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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278
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Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
- 2,095
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59
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Every disk in $(S^2 \times S^1) \setminus B$ whose boundary lies in $\partial B$ separates
Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ ...
- 2,095
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41
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Characterizing centralizer of nilpotent self-maps
Let $\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(...
- 109
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62
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Nontrivial interpretation of dependent type theory in the category of chain complexes
A category $\mathrm{Ch}(\mathbf{Ab})$ of chain complex has a model category structure, which makes the category to interpret dependent type theory.
E.g. a term of a type $A$ is interpreted as an arrow ...
- 101
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215
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Fundamental ring of a circle
Starting with fundamental group, say of a circle, let's reflect back to path groupoid a little. The path concatenation operation is partial, but this can be remedied by focusing on the sets of paths, ...
- 109
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119
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Nullity of the linking matrix of a framed link $L$ equals the first betti number of the manifold obtained by surgery on $L$
I have asked this on mathstackexchange as well. I'm not necessarily asking for a proof, just a hint or a point to the right direction (although I'm not saying that a proof isn't welcome). I'm studying ...
- 101
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89
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relationship between "linear approximation" to immersions and formal immersions
I'm reading these notes
Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$
If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element ...
- 131
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93
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A semifield of characteristic zero may have a finite number of elements
A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.
I ...
- 203
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189
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Grothendieck decomposition of locally free sheaves
Let $f:\mathbb{P}^1 \rightarrow X$ be a morphism with $X$ a smooth projective algebraic variety of dimension $n$, then by Grothendieck $\mathcal{N}_{f}=\bigoplus_{i=1}^{n-1}\mathcal{O}_{\mathbb{P}^1}(...
- 25
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227
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Tangent bundle of symmetric product of surface
Hello everybody please help me with this doubt:
Let $f:\mathbb{P}^{1} \rightarrow \mathrm{Sym}^{d}(X)$, where $X$ is a smooth projective surface, $\mathbb{P}^{1}$ is a projective line and $\mathrm{...
- 25
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160
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Splitting of Atiyah-Hirzebruch Spectral Sequence
Suppse E is a cohomology theory which has Kunneth Formula, i.e $ E(A \wedge B)= E(A) \otimes_{E(pt)} E(B) $. What happens to the Atiyah Hirzebruch Spectral sequence while we compute $ E(A \wedge B) $?...
- 341
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181
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Topological vs algebraic intersection forms
Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...
- 23.3k
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70
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Contractable and Simply Connected Doubling Spaces Homeomorphic to Euclidean Space
Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?
- 5,405
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1
answer
84
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Primage structures: induced domain partitioning by itterated inverse (reference request)
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example, the j-th such preimage list ...
- 23
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214
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Weil cohomology for non-projective varieties
If you have a smooth projective variety over a field and a Weil cohomology theory, weak Lefschetz gives you some cohomological information about its smooth hyperplane sections. Now, if I give you a ...
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1
answer
123
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A characterization of maps that are homotopic relative to $A$ over $S$
Let $$\begin{array}{ccccccccc}
A & \rightarrow & X \\
i\downarrow & & \downarrow p \\
B & \xrightarrow{v} & S
\end{array} $$ be a commutative diagram of simplicial sets, with ...
- 765
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93
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Lifting action to the sphere
Consider the free action of $\mathbb{Z}_2$ on $\mathbb{C}P^n$( $n$ is odd ) by $$[z_0,\dots, z_n]\rightarrow [-\overline z_1,\overline z_0,\dots,-\overline z_{n-1},\overline z_n].$$ Consider the orbit ...
- 395
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213
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make me idempotent
$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$.
$D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$.
$E(D_r)$ is the set of all idempotents of semigroup $T_n$.
$support(\alpha)=\{...
- 109
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94
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$ch(L f^*\epsilon)$
I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,
$ch(f^* \epsilon)=f^* ch(\epsilon)$.
But if $f$ ...
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136
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Amalgamated free-product of semigroups (definition)
I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...
- 233
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248
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A possible generalization of "Group Cohomolgy"
The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor:
$$FIX: \mathcal{M_G} \to \mathcal{Ab}$$
where $FIX$ is the functor from the ...
- 356
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53
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a generalization of group (monoid with order-by-order invertible elements)
Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
- 2,232
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92
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$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]
Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space,
but the statement is easier if we also suppose $X$ ...
- 3,051
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0
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116
views
Open subsets of the n-torus containing no nontrivial loops
Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is ...
user83520
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0
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132
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Small simplicial model for Moore space
I seek for some small simplicial model of Moore spaces $M(\mathbb{Z}/k\mathbb{Z},n)$. What is the simplest way to construct such? Of course we may take some cofiber of degree $k$ map between spheres, ...
- 1,129
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0
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77
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induced homomorphism of adjoint action on fundamental group
Let $G$ be a non-connected compact Lie group and $Ad_g: G \rightarrow G, x \rightarrow gxg^{-1}$ be the adjoint action. Look at the induced homomorphism $ (Ad_g)_* : \pi_1 (G) \rightarrow \pi_1(G)$. ...
- 606
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0
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308
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Generalising the Mayer-Vietoris principle
My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \...
user81684
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0
answers
101
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Spherical Rings
My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse ...
- 501