# Tagged Questions

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

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### mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
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### Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
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### Compact surface homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$

Let $p$ be a point $\in \mathbb{CP}^2$. Is there a compact surface which is homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$ ? I know the homology of $\mathbb{CP}^2$, but I'm not sure about the ...
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### Does there always exist a lift of a path from $Y$ to $X$ if $f: X\to Y$ is a continuous surjective function?

If one has a continuous surjective function $f:X \longrightarrow Y$ and let $\gamma$ be a continuous path in $Y$, under what circumstances can one find a (possibly non-unique) lifted path $\gamma'$ in ...
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### Relation between cup product and intersection number

Suppose $M$ is an oriented diff. manifold and $X$ and $Y$ are two submanifolds of codimension $m$ and $n$ in $M$. Then under some conditions one can define the intersection of $X$ and $Y$ and this is ...
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### deformation retraction for a simple Hopf link

I would like to find a deformation retraction $F:X\times[0,1]\to X$ of the complement of the Hopf link which consists of two circles linked together once. Since Hatcher describes a deformation ...
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### cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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### Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath. One of the questions seems like it should ...
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### What does $d_{n+1}\circ{d_{n}}=0$ mean in the definition of a chain complex?

According to the Wiki article on chain complexes, a chain complex $(A_{\bullet},d_{\bullet})$ is a sequence of abelian groups or modules connected by homomorphisms such that $d_{n+1}\circ{d_{n}}=0$. ...
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### tensor product of trivial line bundles

In Hatcher's book on Vector Bundles, the tensor product of two vector bundles is defined through the gluing functions. But I need an example to understand it. So I think of the simplest case, the ...
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### Mobius band does not retract to boundary circle - specific part

(I'm trying to ask this in a way such that it isn't a duplicate question) The proofs that I've seen for the fact that there is no retraction from the Mobius band to its boundary circle usually say ...
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### Smash product and tensor product of groups

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). ...
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### Proving $S^{4}/G$ is simply connected where $G$ is not a free group action

Consider the sphere $S^{4}$ as a subset of $\mathbb{R}^{5}$ and consider the action of the group $G$ of homeomorphisms generated by $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$ ...
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### Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
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### Dual of path object

For a topological space $X$, what might be the dual of the path space $X^I$ of $X$? Does it make any sense to think of the topological cylinder $X \times I$ over $X$ as dual to the path space over $X$?...
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### CW approximation of $n$-connected space

I want to prove the following lemma: Let $X$ be a n-connected space. Then there exists a CW-approximation $f:K\rightarrow X$ such that $K$ has trivial n-skeleton. What I have done so far: By ...
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### Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
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### Obtaining the Möbius strip as a quotient of $S^1\times[-1,1]$

I am trying to obtain the Möbius strip as a quotient of $S^1\times[-1,1]$, where $S^1$ is, of course, the circle. My definition of Möbius strip is the quotient of the square $[0,1]\times[0,1]$ by the ...
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### Surjection of the fundamental group of a manifold onto a free group induces a map onto a wedge of circles, why?

Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ ...
### The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$
So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
### $f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$
Let $f:S^{2n}\rightarrow S^{2n}$ continuous. Then there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$. I am having a hard time finding a starting Point. Thank you