Happened to be I needed a theorem stating that any 2-dimensional regular CW complex having the property that each of its 1-cells is attached to even number of 2-cells can be decomposed into 2-cell-disjoint spheres. This is the analogue of the true statement, that "an undirected graph can be decomposed into edge-disjoint cycles if all of its vertices have even degree", in one more dimensions. I feel that this is also true, and can be proved similarly, but I didn't find such a statement anywhere.
The proof I know in the case of graphs (1-dimensional CW complexes) is based on the existence of the Euler tour. I think, that Euler tour can be generalized from graphs (1-dimensional CW-complexes) to 2-dimensional regular CW complexes as a tiling process (successive selection of 1-cells and 2-cells) of the 2-complex. In more detail:
- Instead of the vertices of the graph take the 1-cells of the 2-complex.
- Instead of stepping out from a vertex onto an edge, select a 2-cell attached to the given 1-cell.
- Instead of stepping onto a vertex from an edge, select a 1-cell attached to the lastly selected 2-cell.
- The Eulerian property of the walk is that we select every 2-cell exactly once.
- The closedness of the tour means that each 1-cell is attached to even number of selected 2-cells.
I think, that the existence of an Euler tour (here: tiling) can be proved in the same way as in the 1-dimensional case (see here).
Is it OK?