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I know "attaching a handle" to a manifold, but recently I faced "attaching a cell" and I don't know its definition in precise. It seems that the definition is very trivial (!) because my searches did not result in much about the definition, and they were included its application. Can someone help me?

After below answer:

Let $M_0$ be an $n$-dimensional manifold with boundary and let $$φ:S^{k−1}×D^{n−k}→M_0$$ be a smooth embedding such that image $φ$ is contained in $∂M_0$. Define an equivalence relation on the disjoint union $$M_0∐D^k×D^{n−k}$$ by $x∼φ(x)∀x∈S^{k−1}×D^{n−k}$. The quotient space $M_1$ is said to be obtained from $M_0$ by attaching a handle of index $k$.

Now by the definition, I understood that attaching a cell simply is attaching an $n$-handle. Is it correct?

If it is correct, what is the meaning of attaching an $i$-cell?

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Have you looked up the definition of a CW-complex? The definition of "attaching a cell" is part of that definition. You attach cells to arbitrary spaces, not just manifolds. Handle decompositions are the adaptation of CW-structures into the smooth manifold theory world. So from a historical perspective you're learning this topic in reverse. – Ryan Budney Aug 18 '13 at 6:31

You take a ball $D^n$ with a map $f\colon \partial D^n=S^{n-1}\to X$. Then you take the quotient $X\cup D^n/\sim$, where you identify $f(t)\sim t$ for each $t\in S^{n-1}$.

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I think the map should go from $S^{n-1}$ to $X$. – William Aug 17 '13 at 19:17
Yes, you are right. I am sorry. – Sasha Patotski Aug 17 '13 at 19:18

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