# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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### A lemma from Hilton & Stammbach's book A Course in Homological Algebra

In orde to prove the set of equivalence classes of extensions of $A$ by $B$ is a contravariant functor of the first component and covariant functor of the second. The authors give us three lemma. I'm ...
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### Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where ...
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### Is there a nontrivial fiber or principal bundle over $S^3$?

Is there a nontrivial fiber or principal bundle over $S^3$?I know that, by a paper of Steenrod,see the link below, every sphere bundle on 3- sphere is trivial but what about arbitrary fiber ...
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### Do path homotopy classes of concatenated paths have a middle fixed point?

If $[a]$ and $[b]$ are path homotopy classes, then $[a]\cdot[b]$ is defined as $[a*b]$, where $a*b$ is defined as the concatenation of the paths $a$ and $b$. Let us say $a(1)=b(0)=p$. Then does each ...
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### The configuration space of directed great circles on the sphere

I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out. First, he thinks ...
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### Problem 10 of Section 1.2 from Hatcher.

Problem. Consider two arcs $\alpha$ and $\beta$ embedded in $D^2\times I$ as shown in the figure. The loop $\gamma$ is obviously nullhomotopic in $D^2\times I$, but show that there is no ...
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### Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
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### Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
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### Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
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### strange implication of addivity axiom in homology theory

Let $H$ be a homology theory satisfying Eilenberg-Steenrod axioms and $X$ an arbitrary topological space. We can write $X$ as a disjoint union of its points $$X= \coprod_{x \in X}{\{x\}}$$ Now the ...
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### Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
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### Fundamental Property of Regular CW Complexes

$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$. A CW complex ...
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### multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $\circ$ on ...
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### Are the path connected components of $\Omega S_1$ contractible?

Let $\Omega S_1$ be the space of loops of $S_1$ based at $x_0 \in S_1$ with the topology of uniform convergence. We know that the path connected components of this space are in one to one ...
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### Long exact sequence of $(I \times Y, \partial I \times Y)$.

There's a section in chapter 3 in Hatcher's Algebraic Topology where he talks about the long exact sequence of the pair $(Y \times I, Y \times \partial I)$, where $Y$ is any topological space. The ...
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### Showing $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision

I want to show $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision and can't quite figure out how to choose my subspaces. For $Z \subset A \subset X$, excision gives ...
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### Non-abelian fundamental group on a path-connected space

I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise ...
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### How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3;$$ $$H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2};$$  ...
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### universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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### Requirements too stringent for singleton homotopy class [X,Y]?

I recently had a problem: Show that if $X$ is contractible, and $Y$ is path-connected, show that the homotopy class $[X,Y]$ has a single element. I have been able to prove this (I think) in a fairly ...
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### Is there any result on the “counting” of minimal atlas?

Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for$M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is ...
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### $H^*(X,A;R)\cong H^*(X',A';R)\; \Rightarrow H^*(X\times Y ,A\times Y;R)\cong H^*(X'\times Y,A'\times Y;R)?$

I have a quastion about product spaces in singular cohomology. I only know a formula for sinugular homology for product spaces from lecture, the universal coefficient theorem. Let $R$ be a ...
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### A fibre bundle of n-tuples of vectors

The question is: does the surjective, multilinear map $\pi: (\mathbb R^\infty)^n\to \Lambda^n(\mathbb R^\infty)$ defined by $(v_1,\cdots,v_n)\mapsto v_1\wedge \cdots \wedge v_n,$ define a fibre ...
### Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.
Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...