Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $R = {\bf Z}$

Let $\partial_i : C_i \rightarrow C_{i-1}$ be a boundary map

where $C_{-1} = \{ 0 \}$, $C_i$ is the set of all maps $f$ and $f: \Delta_i \rightarrow M$. Let $Z_n = $Ker of $\partial_n$

If $f $ is a loop in $M$ then $\partial_1(f)=0$.

If $M$ is $S^2$, and if $f$ is in $Z_2$, what is $f$ ?

I want to know the example for $f$.

In fact, if $f$ maps the boundary of $\Delta_2$ into a point in $M$, i.e., $\Delta_2$ wraps $M $ one time, $f$ is in $Z_2$. However does it imply that $\partial_2 (f) =0$ ?

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

If $f$ is as you describe, and $1,2,3$ denote the three endpoints of $\Delta_2$ then $\partial_2 f = f_{23} - f_{13} + f_{12} = \ast - \ast + \ast = \ast$ where $\ast \in M$ is the point you map the boundary to and $f_{ij}$ is $f$ restricted to the boundary path from $i$ to $j$. Hence the answer is no, in this case $f$ is not an element of $\mathrm{ker} \partial_2$.

Note that your notation is not standard. Commonly, $C_i$ is used to denote the $i$-th chain group which is the set of all formal sums of $i$-simplexes. Hence if $\sigma_i : \Delta_i \to M$ is a continuous map then an element of $C_i$ looks like $\sum_{k=1}^n c_k \sigma_k $ for $c_k \in \mathbb Z$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.