Nerve Lemmas

In Euclidean space, it can be seen that the weighted Voronoi regions and all their intersections are contractible therefore it follows by the nerve lemma (Theorem 15.21) that the dual complex of a ball collection is homotopy equivalent to the union of these balls. 

Remark 15.22. If X is a CW complex, the nerve lemma holds for covers by subcomplexes. 

Proof of nerve lemma. Let T> be the nerve diagram of tl, and let D be the trivial diagram over BdA/"(W), i.e., all associated spaces are points. Consider the unique diagram map X -.V V. Since all maps X cr) for a Af(U) are homotopy equivalences, we can conclude by the homotopy lemma that also the map hocolimX hocolimP — hocolimH is a homotopy equivalence. 

Furthermore, by the projection lemma, the map pj hocolimP —> colimP is a homotopy equivalence. Since coIimP = X and hocolimP = N IA), the nerve lemma follows. ... 

As an application of the nerve lemma, let us prove the topological equivalence of the two product notions that we have so far defined for abstract simplicial complexes. 

Sometimes, we do not know that the covering complexes and their intersections are contractible. Instead, only partial information about their connectivity can be extracted. The more general nerve lemma then allows us to draw a weaker assumption about the connectivity of the total space. 

The thin (8 nm) outer cell membrane or "plasma-lemma" (Fig. 1-7) controls the flow of materials into and out of cells, conducts impulses in nerve cells and along muscle fibrils, and participates in chemical communication with other cells. Deep infoldings of the outer membrane sometimes nm into the cytoplasm. An example, is the "T system" of tubules which functions in excitation of muscle contraction (Figs. 19-7, 19-21). Surfaces of cells designated to secrete materials or to absorb substances from the surrounding fluid, such as the cells lining kidney tubules and pancreatic secretory cells, are often covered with very fine projections or microvilli which greatly increase the surface area. 

Proof of the gluing lemma. lfXnB = CnB = 0, then the statement is trivial, so assume that these are nonempty. Let T>i be the nerve diagram... 

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