Cellular Homology, Relative Homology of Skeletons

The Homology of a Closed Cell Relative to its Boundary

Given a cell E, let f be its characteristic function, let E_c be the closure of E, and let E_b be E_c-E (the boundary). Remember that f acts on Dⁿ, with boundary S^n-1 mapping onto E_b.

Let V be a small closed ball inside Dⁿ, E.g. the points with norm ≤ ½. Let W be the image of V under f. Since f is a homeomorphism inside Dⁿ, W is closed inside E.

Build a 2 by 3 grid of relative singular homology groups, with functions heading towards the middle and down. The vertical functions are defined by f.

The arrows from right to left are inclusion maps. Start at the top right and move to the middle, thus embedding a ball without its surface into the same ball with its surface. In each case we are interested in the relative homology, with and without V. apply the excision theorem, thus removing the surface, and the embedding induces an isomorphism on relative homology.

The same thing happens downstairs, as E is embedded into E_c. Excise E_b to demonstrate the isomorphism.

We will demonstrate homology isomorphisms from left to middle by building a homotopy equivalence. The arrow from left to middle is the identity map. The reverse map expands V outward, and squashes everything outside of V radially onto the surface. This is also the composition of the two maps in either order. A homotopy pulls the disk back towards V and recreates the layer between V and the surface. This brings us back to the identity map. Put this all together and find a homotopy equivalence, which induces an isomorphism on relative homology.

The same thing happens downstairs, though the proof is a bit more technical, because you have to push everything through f first. The arrow is the identity, as it was above. To go backwards, reverse the action of f, expand V out to the surface, and apply f. This is well defined everywhere except E_b, because f is a homeomorphism inside - and there is no trouble with E_b, because that is fixed in place. The homotopy is built the same way, with E_b fixed throughout time. I'll leave the detailed δ ε proof of continuity to you. Therefore all the horizontal arrows induce homology isomorphisms.

Since the horizontal arrows are inclusions, and the vertical arrows are defined by f across all of Dⁿ, the diagram commutes.

The arrow down the right side is an isomorphism, because f is a homeomorphism on the interior of Dⁿ. That means all three vertical arrows are isomorphisms.

By an earlier theorem, the homology of the disk relative to its boundary is known. The result is Z for h_n, and 0 in all other dimensions. The same holds for the homology of E_c relative to E_b.

All Together Now

Let U be the disjoint union of balls of dimension n, one ball for each cell of dimension n in S. In our example, U is two open disks that map homeomorphically onto the lenses. Let f be all the characteristic functions of the n dimensional cells running in parallel. Thus f is a function from U into S.

A closed set in S is closed in each closed cell. This is certainly true for the closed cells of n dimensions. Pull back through f and find closed preimages in each n ball. This is a closed set in U. Therefore f is continuous. Like any continuous function, it induces a homomorphism on homology and relative homology.

Let E be an open n dimensional cell in S, and restrict attention to the n skeleton of S. Let T be the complement of E. Let C be a cell in T. The boundary of C is covered by cells of lower dimension, hence the closure of C is disjoint from E. The closure of C is entirely in T, and T intersects the closure of C in the closure of C, which is closed. At the same time, T intersects the closure of E in the boundary of E, which is closed inside the closure of E. Therefore T intersects every closed cell in a closed set, and T is closed. Since E is arbitrary, every n dimensional open cell in S is open in S.

The union of the n cells is a disjoint union of open sets in the n skeleton of S. Each of these is homeomorphic to the interior of its corresponding n ball. Put this all together and find a homeomorphism between the disjoint union of the interiors of the balls of U and the open n cells of S. The right side of our 2 by 3 grid, when applied to all the n cells in parallel, remains a homology isomorphism.

As a side note, the boundary of an n dimensional cell E is closed. We never proved this before, let's show it now. The closure of E is closed, of course. Intersect this with the complement of E, which is also closed, and find the boundary. Thus the boundary of E is closed in the n skeleton. But the n skeleton is closed in S, hence the boundary of every cell is closed in S. The boundary of a point is the empty set, so this too is closed in S.

Now, back to the commutative diagram. The arrows from right to middle rely on the excision theorem, which means the closure of the space being excised must live within the interior of the relative space. As shown by an earlier theorem, the n-1 skeleton is closed. In other words, the frames are closed in the glasses. We only need show the glasses, with holes in the lenses, form an open set.

It is equivalent to prove the collection of holes is closed. Intersect with each closed cell and you get nothing, or the image W of V in an n dimensional cell E. Since the closed ball is compact, and S is hausdorff, the continuous function f from the closed ball onto the closure of E becomes bicontinuous. Therefore the image W of the closed set V is closed. Closed in every closed cell means the collection of images W_i is closed, and the complement is indeed open. We can safely use the excision theorem to assert homology isomorphisms from right to middle.

The arrows from left to middle become isomorphisms thanks to a homotopy equivalence that applies to all the balls in parallel. Each V pushes out towards the surface of its ball, and retracts back again. The same thing happens downstairs, in all the n dimensional cells simultaneously. I'll leave the details to you.

The down arrows are the function f, and the diagram commutes, as it did before. Therefore the left arrow is an isomorphism. The homology of the n skeleton, relative to the n-1 skeleton, is isomorphic to another relative homology, based on balls and their boundaries. This is a disjoint union of spaces, so we can consider each ball separately, and then put it all together. Therefore the relative homology is free abelian, with one generator for each n dimensional cell.

Relative Homology and the Quotient Map

Let V be a closed set in S, and let U be an open set in S that retracts to V. Let T be the quotient space of S that identifies all of V with a point w. Note that U is still an open set in T; but the subspace V has been replaced with the point w.

Let h be the homotopy from S into S that squashes U down onto V. This is a retract, hence h fixes V throughout time. Apply h to T. Note that h fixes w throughout time, and squashes U down onto w. Combine the properties of a homotopy and a quotient space to prove h is continuous, whence w is a retract of U within T.

Build a 2 by 3 commutative diagram, as we did above.

Since the quotient map fixes everything outside of V, the rightmost vertical arrow is a homeomorphism. In other words, S-V and T-w are the same space.

Consider the arrows from right to middle. Remember that V, or w, is closed inside the open set U. Apply the excision theorem, and find a relative homology isomorphism.

The arrow from left to middle is the identity map, as it was before. A reverse arrow, from middle to left, squashes U onto V. The composition, in either order, gives the function that squashes U onto V, which is homotopic to the identity map. Note that the image of V lies in U, and the image of U lies in V, and the homotopy keeps U within U, and V within V. We have another homotopy equivalence, inducing an isomorphism on relative homology.

Put this all together and the three vertical arrows are isomorphisms. If V is a retract of some open set U, the quotient map that compresses V down to a point w preserves the homology of S relative to U, or V|w.