$\quad$ The functor $H_n$ measures the number of “$n$-dimensional holes” in the space (or simplicial complex), in the sense that the $n$-sphere $S^n$ has exactly one $n$-dimensional hole and no $m$-dimensional holes if $m\neq n$. A $0$-dimensional hole is a pair of points in different path components, and so $H_0$ measures path connectedness. The functors $H_n$ measure higher dimensional connectedness, and some of the applications of homology are to prove higher dimensional analogues of results obtainable in low dimensions by using connectedness considerations.

Can someone explain me what it means ?

if i understand $H_n$ measure the numbers of holes with dimension $n$ but what about $H_0$ what is the relation between the holes of dimension $0$ with path connected component ?

Thank you.


Loosely, $H_0 (X) \cong \mathbb{Z}\oplus \mathbb{Z}$ means that there are two path components. Think of a curve. If there are two points which have no curve connecting them, we can pretend like there is a zero dimensional hole in the curve that would connect them. So, if $H_0 (X) \cong \mathbb{Z}$, we know that $X$ is connected.

  • $\begingroup$ ok, please what is a n-dimensional holes , the definition ? $\endgroup$
    – Vrouvrou
    Mar 27 '14 at 7:17
  • 2
    $\begingroup$ There is no definition. It is just for your intuition. $\endgroup$
    – Zhen Lin
    Mar 27 '14 at 8:27
  • $\begingroup$ n-dimensional holes is a n-cycle ? $\endgroup$
    – Vrouvrou
    Mar 27 '14 at 11:06
  • $\begingroup$ zero dimensional holes are weird in that they are just the number of components. One dimensional holes are something that gives you a loop that you cannot homotopy to the constant loop. So, a circle works. If you would puncture the plane, $\mathbb{R}^2$ three times, that is the same as having the bouquet of three circles. $\endgroup$
    – N. Owad
    Mar 27 '14 at 15:40
  • 1
    $\begingroup$ There is not a definition of an "n-dimensional hole". It's just something you can 'see' that homology measures; it's mentioned so that one can build intuition for homology. Hatcher's "Algebraic Topology" is a good place to learn the subject $\endgroup$
    – user98602
    Mar 27 '14 at 20:26

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