Hatcher 0.8

Hatcher, Algebraic Topology, Chapter 0

8. For n>2, construct an n-room analog of the house with two rooms.

Proof. First, consider Dr. Hatcher’s description of the house with two rooms, shown in Figure 1 below.

To build this space, start with a box divided into two chambers by a horizontal rectangle, where by a ‘rectangle’ we mean not just the four edges of a rectangle but also its interior. Access to the two chambers from outside the box is provided by two vertical tunnels. The upper tunnel is made by punching out a square from the top of the box and another square directly below it from the middle horizontal rectangle, then inserting four vertical rectangles, the walls of the tunnel. This tunnel allows entry to the lower chamber from outside the box. The lower tunnel is formed in similar fashion, providing entry to the upper chamber. Finally, two vertical rectangles are inserted to form ‘support walls’ for the two tunnels. The resulting space X thus consists of three horizontal pieces homeomorphic to annuli plus all the vertical rectangles that form the walls of the two chambers. – Allen Hatcher’s Algebraic Topology, pg 4

Hatcher 0.8

Figure 1
The house with two rooms

The most obvious extension of this construct for n>2 would seem to be as follows: Take a box B with no top and no bottom and divide it into n chambers by n-1 horizontal (interior) rectangles R_i, i=1,\ldots,n-1, where – without loss of generality – we assume the dividing rectangles R_i are equally spaced among the height of the original box B. For convenience, assume the numbering of the R_i increases from bottom to top. Next, close B by attaching a “floor rectangle” R_0 and a “roof rectangle” R_n, resulting in a closed box with n layers constructed by way of (n-1)+2=n+1 horizontal rectangles.

Next we create a tunnel using a sort of induction. As in Dr. Hatcher’s construction, create two tunnels, one going from the “roof” R_n to the “floor” R_{n-1} of the n^{\text{th}} “story” and the other going from the “floor” R_0 to the “roof” R_1 of the first “story”. Insert the vertical support walls as needed. If these two tunnels end on the same R_i, then connect them and stop. Otherwise, create the “intermediate” tunnel(s) by (without loss) a bottom-up construction where R_1 is connected to R_2 by a tunnel + support walls, R_2 to R_3, etc., until the sequence of intermediate tunnels ends on R_{i-1}.

When this happens, connect the last intermediate tunnel to the tunnel connecting R_n to R_{n-1} and stop. The result is analogous to the construction given above and consists of n rooms, one tunnel in each, built from a box with n+1 horizontal rectangles.   \square

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