Fundamental group of CW complexes

algebraic-topologycw-complexesfundamental-groups

I was building a small text on the fundamental group of CW complexes. I followed https://www.homepages.ucl.ac.uk/~ucahjde/tg/html/vkt02.html, where I have the result that the fundamental group of a CW complex depends only on its $1$-cells and how the $2$-cells are attached.

But there is something not so clear about this, for example: If I construct the solid torus $\mathbf T$ as

  • one $0$-cell, $v$
  • two $1$-cells, $a$ and $b$ whose endpoints are attached to the $0$-cell
  • one $2$-cell, $A$ attached along the loop $aba^{-1}b^{-1}$
  • one $3$-cell, $S$ attached to the the $2$-cell $A$ (is this possible?) such that it fills the torus

Then I get that $\pi_1(\mathbf T)\simeq\mathbb Z\times \mathbb Z$, but this is wrong since one of the generating loops is now homotopic (through the $3$-cell) to the constant loop, and so $\pi_1(\mathbf T)\simeq\mathbb Z$.

Makes me wonder if this result about the dependence of $\pi_1(X)$ (for $X$ a CW complex) only applies for $k$-cells that attach to the $1$-cells; and that it fails to be true about $k$-cells that are attached to $2,3,4,\dots$-cells.

Can anyone shine some light on this?

PS: I did not study amalgamated relations for this, I only went with the material used in Hatcher's Algebraic Topology https://pi.math.cornell.edu/~hatcher/AT/AT.pdf, for example in Proposition 1.26, page 50, where he uses the quotient definition of presentation.

In Proposition 1.26 point (a), after a short paragraph describing that the attaching of cells occurs through an attaching map $\varphi_\alpha:S^1 \to X$, Hatcher says,

If $Y$ is obtained from $X$ by attaching $2$-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection $\dots$

Which further leads me to believe that it only applies, like I said before, to $k$-cells attached along $1$-cells.

Best Answer

You can't attach just a single 3-cell to fill in the torus, since the inside of a torus is not homeomorphic to a 3-dimensional ball. Instead you should first fill in the "inside" of the second 1-cell with a disk, and then you can glue a 3-cell. This second 2-cell kills off one of the generators.

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