# Tagged Questions

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### How does one compute a group modulo a torsion group?

Let's say I have some group $G$ and a subgroup $H$ such that $H$ is a torsion group (i.e. $\forall h \in H$, $h$ has finite order. How do I compute the factor group $\frac{G}{H}$? What effect does the ...
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### An Advice Concerning Master's Programme [on hold]

Which of these programmes is a better choice, if one wants to pursue a degree in pure mathematics? (In Geometry, Topology and Algebra, in particular, algebraic geometry) 1) ...
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### Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X =$ (the Hawaiian Earring) which ...
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### Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
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### homeomorphism classes of compact surfaces with addition operation is a monoid

This is essentially pg 6 of serge lang's algebra's discussion about an interesting example. Homeomorphism classes of compact surfaces with the addition operation defined as following. Say M and $M'$ ...
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### map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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### How to prove a direct sum?

$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint. $i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by ...
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### Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)