3
$\begingroup$

Let $m \in \mathbb{N}$. Can we have a CW complex $X$ of dimension at most $n+1$ such that $\tilde{H_i}(X)$ is $\mathbb{Z}/m\mathbb{Z}$ for $i =n$ and zero otherwise?

  • 0
    Yes. Hint: You know how to do this for $\mathbb{Z}$, now try to topologically realize the exact sequence $\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z}/m$ using a cofiber sequence.2012-12-12
  • 0
    I know the answer for $m = 1$. What is the meaning of cofiber sequence?2012-12-12
  • 0
    You don't need to know what a cofiber sequence is to answer your question. The idea is to take a sphere of dimension $n$ and attach a single $n+1$-cell. You need to know that homotopy-classes of maps $S^n \to S^n$ are in bijective correspondence with the integers (called degree) and using the degree $m$ map gives you what you want.2012-12-12
  • 1
    OK I understood now. I have to attach $D^{n+1}$ to $S^n$ via a degree $m$ map. So the resulting space will consist of one zero cell, one $n$ cell and one $n+1$ cell. Then I have to compute cellular homology2012-12-12

1 Answers 1