Questions tagged [algebraic-topology]
Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.
20,682
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Cohomology ring of symmetric products (of manifolds)
Let $S_g$ be a closed, orientable surface of genus $g$ (new notation in light of the first comment). I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring ...
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0
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22
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How to compute the monomial matrices?
In GTM227 Combinatorial Commutative Algebra, Miller defined the monomial matrix to represent a map between two $\mathbb{N}^n$-graded free $S$ modules. Its columns are labeled by sourse degrees $a_p$ ...
- 136
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1
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Examples of CW-complexes wich are 1-acyclic but no simply conected.
Hurewicz theorem states that if $X$ is a simply-connected CW-complex then $X$ is $(n-1)$-connected if and only if it is $(n-1)$-acyclic and that in this case $\pi_n(X)=H_n(X)$. Moreover, it is also ...
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A tricky question on $O(2)$ bundles over surfaces
Let $\Sigma = \mathbb {RP}^2$, and consider a non-orientable $O(2)$-principal bundle $\pi:P\to \Sigma$, this means that $w_1(P)\neq 0$, the first Steifel-Whitney class.
Form the associated bundle $E= ...
- 5,488
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7
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Intersection of subspace of cyclical rotations with orthant
In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector
$\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
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Construct cocyle from Leray-Serre spectral sequence
Let $F \xrightarrow{f} E \rightarrow B$ be a fiber bundle and $L$ is the locally constant sheaf on $E$. $B$ is NOT simply connected.
We can apply Leray-Serre spectral sequence to compute the local ...
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Uniqueness of comultiplication for sufficiently connected spaces of restricted dimensions
I'm working through an introductory homotopy theory book (Arkowitz) for self-study, and I'm a bit stuck on the following exercise. Not a lot of machinery is available at this point in the book. I'm ...
- 436
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Prove that any connected cover of $S^1$ of finite degree is homeomorphic to $S^1.$
Assume that any connected and compact $1$-manifold is homeomorphic to either the unit circle $S^1$ or the unit interval $[0, 1].$ Prove that any connected cover of $S^1$ of finite degree is ...
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Warsaw circle has the fixed point property
I'm looking for any hint on proving that the Warsaw circle has the fixed point property. By this I mean that for every continuous function $f$ from the Warsaw circle onto itself there exists a point $...
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The complementaries of homologous objects in $S^n$ are homologous?
Suppose $A$ and $B$ are two subsets of the $n$-sphere $S^n$ that have isomorphic finitely-generated homology groups. Do $S^n\setminus A$ and $S^n\setminus B$ also have isomorphic homology groups?
I ...
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1
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Contractibility of the based path space.
Let $X$ be a topological space and $x_0 \in X.$ Consider the based path space $$\mathcal P_{x_0} (X) : = \left \{\gamma : [0,1] \xrightarrow{\text {continuous}} X\ \bigg |\ \gamma (0) = x_0 \right \}$$...
- 343
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Spectral sequence for a truncated semi cosimplicial space
Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
- 6,098
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How to draw Caley complex for any group
I am reading Algebraic Topology by Allen Hatcher. I come to know
for any group $G$ ,we can make an universal cover $X$ of $X/G$ by properly discontinuous action .
There is a paragraph mentioned in ...
- 1,162
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Relationship between compactly supported homology and homology with orientation sheaf
Let $M$ be noncompact and non-orientable manifold of dimension d. Is it true that $H_{c}^{d-i}(M;Q)$ and $H^{i}(M;Q^{w})$ are isomorphic, where $Q^{w}$ is orientation sheaf and $Q$ is a field of ...
- 1,026
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Topology of Mobius Strip
The mobius strip $M$ is topologically distinct from the cylinder $S^1\times I$ where $I$ is a finite segment of $\mathbb{R}$ (namely, one cannot be deformed into the other without cutting and pasting)....
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If $H$ is a closed subgroup of a Lie Group $G$, and $p:P\to B$ a principal G-bundle. How to show that $q:P\to P/H$ is a principal $H$-bundle?
Let $G$ be a Lie group and $H$ a closed subgroup of $G$. Suppose we are given a principal $G$-bundle $p:P\to B$, how to show that the quotient $q:P\to P/H$ is a principal $H$-bundle? Where $P/H$ ...
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1
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Hatcher 1.1.18 - sufficient solution?
Could someone check the following solution to Hatcher 1.1.18? I'm wondering whether the arguments in the end using van Kampen are sufficient in normal proof-writing in algebraic topology.
Problem:
...
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Show that the map $s : X \longrightarrow \widetilde X$ is well-defined.
Let $p\ \colon \widetilde X \longrightarrow X$ be a covering map such that any loop $\gamma$ in $X$ based at $x_0 \in X$ lifts uniquely to a loop based at $\widetilde {x_0} \in p^{-1} (\{x_0\}).$ ...
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Does path connectedness simplify the proof of the LES in reduced homology?
I'm reading Weintraub's Fundamentals of Algebraic Topology, in which there is an exercise (3.4.6 in the first edition) that wants us to show that for every path connected subspace $A$ of a path ...
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What was the intended, more elementary solution to Hatcher $2B.5$? On isomorphisms in homology between a sphere complement and a subsphere
I solved Hatcher's exercise $2B.5$ but I wonder if there is a more elementary approach. Paraphrasing, and removing trivial or vacuous cases, the exercise is this:
Suppose $n\ge1$ and $0\le k\le n-1$ ...
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1
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Bilinear pairing on homotopy groups
Let $X,Y,Z$ be pointed spaces, and $f:X \wedge Y \rightarrow Z$ a map.
Then, $f$ induces a bilinear pairing $\pi_n(X) \times \pi_m(Y) \rightarrow \pi_{n+m}(Z)$ ($n,m \geq 1$).
I see what the pairing ...
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Mapping cylinder being mobius band - example 1.35 in Hatcher
A lot of the geometric examples in Hatcher with abstract identification spaces I have a hard time visualizing. In the following example, Hatcher says that the mapping cylinder of $z \to z^2$ is the ...
- 135
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1
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The Mackey functor $\underline{\pi}_n(X)$
Let $G$ be a finite group and $X$ a pointed $G$-space.
The assignment $G/H\to \pi_n(X^H)$ should define a Mackey functor. I am trying to figure out what the transfers and restrictions are.
If $H\...
- 1,803
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Is the quotient space of a homogeneous space induced by a free group action s.t. the quotient map has a right inverse homogeneous?
Given is a topological space $X$ and a group $G \leq$ Aut($X$) with the property: for $\lambda \in G$ and $x \in X$, $\lambda(x)=x \Rightarrow \lambda=id_X$, also satisfying that the quotient map $q:X ...
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Adapting a deformation retraction onto an adjunction space
I am struggling to understand part of the top answer here: Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell
A space $A$ is glued onto a general topological space $X$ ...
- 4,613
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A cell complex homeomorphic to $S^n$
Proposition. If $X$ is a finite $n$-dimensional cell complex, such that each $(n-1)$-cell is contained in exactly two $n$-cells, then $X$ is homeomorphic to $S^n$.
This statement sounds quite ...
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2
answers
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Pushout of equivalences along cofibrations are equivalences
I would like to show that if $i:A\to X$ is a cofibration and $f:A\to B$ is a homotopy equivalence, then the induced map $k:X\to X\cup_AB$ is again a homotopy equivalence.
$\require{AMScd}$
$$
\begin{...
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How can one prove that circle isn't a retract of a ball without using Homotopy or Homology theory?
Are there some "elementary" ways to prove that circle can't be expressed as a two dimensioanl disks retract?
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Cohomology group with torsion that is not 2-torsion
Are there examples of topological spaces with cohomology/homology groups that have torsion that is not 2-torsion?
- 13
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Mapping between LSQ for several samples and one single sample
I have a sysid problem of the type:
$Y = AX$,
where X is a matrix whose columns are different input vectors, and Y is the output.
Therefore, A is a general matrix that maps all the input vectors(...
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1
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Separating subspheres from each other; what are the weakest hypotheses required in Hatcher exercise $2B.4$?
The task:
Take integers $p,q\ge1$ and define $S^{p-1}\subset S^{p+q-1}\subset\Bbb R^{p+q}$ to be the subsphere consisting of points of $S^{p+q-1}$ whose last $q$ coordinates are zero, and define $S^{...
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Prove that $p \circ p'$ is a covering map
Problem:Let $p:X_2 \rightarrow X_1$ and $p':X_1 \rightarrow X$ be two covering maps.If $X$ is locally path connected and semilocally simply connected the $p\circ p'$ is a covering map.
My attempt: Let ...
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Let Omega = [0; 1] Let the class C consist of sets A such that or |A| < ∞, or |A ^ c| < ∞.
Let Omega = [0; 1] Let the class C consist of sets A such that or |A| < ∞, or |A ^ c| < ∞. Such sets are called cofinite. For example, the set A = {0.1, 0.2, 0.3} is cofinite because it is ...
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Persistence Vector Spaces
I am currently reading Gunnar Carlsson's "Topological Pattern Recognition for Point Cloud Data", you can find it here: http://math.stanford.edu/~gunnar/actanumericathree.pdf
I have a ...
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Find a subgroup $K$ to complete the pullback diagram $G/g_1Hg_1^{-1}\leftarrow G/H\to G/g_2Hg_2^{-1}$.
EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$.
Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite ...
- 1,803
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Sufficient conditions for retractions along simplices to result in "nice" complex
Suppose we have a connected simplicial complex $X$ such that we can partition the vertex set of $X$ into disjoint $M_1,\ldots,M_k$ such that for all $i$, the induced simplicial complex on the vertices ...
- 1,774
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1
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The path components of $Map(X,Y)$ is one to one corresponding to $[X,Y]$, the set of homotopy classes between $X$ and $Y$.
Show that The path components of $Map(X,Y)$, equipped with compact-open topology with a subbasis
$$ \mathcal{O}_{K,U}:=\{f\in Map(X,Y): f(K)\subseteq U\},$$
is one to one corresponding to $[X,Y]$, the ...
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What is the mean of "the isometries that identify the sides of polygon"?
Picture below is from the 167th page of do Carmo's Riemannian Geometry. I don't know the mean of "the isometries that identify the sides of $P$". My English is poor. I know the process of ...
- 5,783
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Nerve Theorems for Open Coverings [closed]
There are numerous Nerve theorems that come down to something like this: For an open cover U of a space X, let N(U) be the nerve. Then under certain conditions, N(U) is homotopy equivalent to X. The ...
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Clarification on the term "divisibility mod torsion" in cohomology
I came across the following statement when reading Section 4 of this paper:
For an oriented 2-plane field $\xi$, let $d(\xi) \in \mathbb{Z}$ denote the divisibility of the Chern class, so that $c_1(\...
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Every co-finite topology is compact?
Has trouble understanding finite set while I was solving theorem on topological space which is
Every co-finite topology is compact.
Let (X,T) be topological space.
In co-finite topology X is infinite ...
4
votes
1
answer
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Prove a space is contractible
I encountered this problem while reading the proof of Lemma 1.2.4.17 in Jacob Lurie's Higher Algebra
Recall that a topological simplex $|\Delta^n|$ can be identified with $\{0\leq x_1\leq\cdots\leq ...
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0
answers
47
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Representation of topological K-theory via Brown representability
We know that topological K-theory is a generalized cohomology theory, and reduced K-theory is a reduced cohomology theory. Thus, both are representable with a sequence of pointed homotopy functors, ...
- 138
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Limitations of Ω-Spectra
Right now I am interested in the "stabilization" endofunctor of the category of ∞-groupoids sending an object $X$ to $\text{colim } Ωⁿ Σⁿ X$. This colimit is related to $∃Y:X≅ΩY$. In Ω-...
- 53
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answers
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Explicitly understanding the map representing first cohomology
Let $M$ be a closed smooth manifold. It is well-known that there is a bijection between the cohomology group $H^n(M;G)$ and $[M,K(G,n)]$, where $G$ is a group and $K(G,n)$ is an Eilenberg-Maclane ...
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1
answer
81
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Spaces with two non-trivial homotopy groups
I'm wondering if there is any elementary example of a space with precisely two non-trivial homotopy groups.
Let $X$ be a connected CW complex with precisely two non-trivial homotopy groups $\pi_p$ and ...
- 2,031
2
votes
0
answers
98
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Hatcher 1.2.4 solution?
Problem 1.2.4: Let $X \subset \mathbb{R}^3$ be the union of $n$ lines through the origin. Compute $\pi_1 (\mathbb{R}^3 - X)$.
I have the following solution. Would someone be so kind as to check ...
- 135
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0
answers
61
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A Problem With Simplicial DeRham Cohomology
I am getting a contradiction by calculating the simplicial DeRham complex of an arbitrary manifold and getting it to be trivial. I also get a similar contradiction using the Godement resolution for ...
- 1,354
3
votes
0
answers
57
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Equivariant Cohomologies of Two Homotopy Equivalent Topological Groups
Suppose that $K \subset G$ is a topological subgroup and this inclusion is a homotopy equivalence (so somewhat stronger than what's written in the title). I'm not assuming compactness but am happy to ...
- 1,725
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1
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40
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Estimating Vectors from the Convex Hull
Let us consider a set of $N$ vectors, $\mathbf{h}_1, \mathbf{h}_2, \cdots, \mathbf{h}_N$, such that $\mathbf{h}_i \in \mathbb{R}^{M}$, $\forall i$, with $M < N$.
Let us also consider the space $\...
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