Answer by user934527 for Meaning of "holes" counted by homology groups
The other answers are good at explaining how to think about the standard theory, but missing some discussion on why this formulation is preferable I think.The punchline for your intuition: It is...
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Answer by ಠ_ಠ for Meaning of "holes" counted by homology groups
Briefly, the dimension of the hole is the dimension of the "witness" needed to detect it.In a bit more detail, when we talk about a $k$-dimensional hole in a space, it just means that you need a...
View ArticleAnswer by pancini for Meaning of "holes" counted by homology groups
I'm not sure if this adds much to the other answer, but here are some thoughts from someone who got into topology fairly recently.First of all, the concept of homology representing $k$-dimensional...
View ArticleAnswer by HallaSurvivor for Meaning of "holes" counted by homology groups
The "dimension" of a hole is the dimension of the part that actually exists. For a sphere, then, we have a $2$-dimensional boundary, which is missing a $3$-dimensional ball inside it. We call that a...
View ArticleMeaning of "holes" counted by homology groups
In a lot of more or less informal introductions to simplicial homologyoften the groups $H_k(X)$ of a topological space or CW spaceare introduced as groups which "counting $k$-dimensional holes". I know...
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