# Tagged Questions

**11**

votes

**0**answers

448 views

### About maps inducing bijections on homotopy classes

Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...

**4**

votes

**2**answers

617 views

### Homology of Covering Spaces

Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...

**12**

votes

**2**answers

592 views

### Are acyclic subcomplexes of finite contractible 2-complexes contractible?

Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi_1(X)$ ...

**22**

votes

**3**answers

963 views

### Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$G_0 > G_1 > ... > ...

**3**

votes

**3**answers

324 views

### Reference needed: Isomorphism on pi_1 and homology gives weak equivalence

Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi_1(X) \to \pi_1(Y)$ is an isomorphism, and $H_n(f) : H_n(X, G) \to H_n(Y, G)$ is an isomorphism for all $n \ge ...