Cw Complexes and Cellular Homology — Examples

The attaching map of the $n$-cell has incidence degree $1+(-1{)}^n$, so Cellular boundary is the incidence degree matrix gives the stated alternating $0,2$ differentials.

Taking kernels modulo images gives the listed groups, including ${\mathbb{R}\mathbb{P}}^0$ where only $C_0=\mathbb{Z}$ occurs; Cellular homology computes singular homology identifies them with singular homology.

There are no adjacent-dimensional cells, so all cellular boundaries vanish by A CW complex with no cells in adjacent dimensions has zero cellular boundary.

The cellular groups are one copy of $G$ in the listed even dimensions and zero otherwise, and the comparison theorem gives the asserted singular groups.

Each oriented edge occurs once positively and once negatively in the abelianization of the word, so every incidence coefficient of the face is zero; the edge-to-vertex boundary is also zero.

The cellular complex is $0\to \mathbb{Z}\to {\mathbb{Z}}^{2g}\to \mathbb{Z}\to 0$ with zero maps, and Cellular homology computes singular homology proves the calculation.

Its incidence degrees are $0,p,0$ in dimensions $3,2,1$, respectively, so the displayed complex follows from Cellular boundary is the incidence degree matrix.

Its kernels and images are the stated groups (also for $p=1$, when $\mathbb{Z}\mathrm{/}p=0$), and the cellular comparison transfers them to singular homology.

The finite skeleton ${\mathbb{R}\mathbb{P}}^m$ has the alternating $0,2$ cellular complex and the resulting singular homology computed in Cellular homology of real projective space. In each fixed degree these groups and the inclusion maps stabilize once $m$ is larger than that degree.

Taking the stabilized groups and applying Homology of an infinite CW complex is the colimit of skeletal homology yields the stated infinite-dimensional calculation.

Their alternating cell counts are $1-1=0$ and $2-2=0$, using Euler characteristic of a finite CW complex.

Both agree with $\mathrm{rank}{H}_0(S^1)-\mathrm{rank}{H}_1(S^1)=1-1$ by Euler–Poincare formula for finite CW complexes.

By Cell attachment by a characteristic map, the closed-cell image is $D^1\mathrm{/}\mathrm{\partial }{D}^1\cong S^1$.

Its open cell is still the embedded open interval by The interior of an attached cell embeds openly in its closure, but $S^1$ is not homeomorphic to $D^1$; this refutes the statement.

Each punctured circle is the image of the interior of a characteristic interval, and its closed circle meets only itself and the common vertex. Thus this is a closure-finite characteristic-cell decomposition.

Put the radius-$1\mathrm{/}n$ circle tangent at the origin with centre $(1\mathrm{/}n,0)$, and choose its far point $x_n=(2\mathrm{/}n,0)$. The set $\{x_n:n\ge 1\}$ meets every closed cell in a closed set, but it is not closed in the Hawaiian earring since $x_n\to 0$. Hence the topology is not the weak topology on these closed cells, so this decomposition is not a CW complex.

The Hawaiian earring is compact as a closed bounded subset of ${\mathbb{R}}^2$, and it meets every proposed open $1$-cell (each punctured circle).

This violates A compact subspace of a CW complex meets only finitely many cells, so the proposed cell structure is not a CW complex.