Understanding the $\Delta$-complex structure of a quotient space

Question

I'm reading Hatcher's book of Algebraic Topology where he define the notion of $\Delta$-complex. Then he puts the following diagram

And said that this shows a $\Delta$-complex structure of the torus, the real projective plane and the Klein bottle. I'm trying to figure out how this structure is. This is my interpretation.

If I have a $\Delta$-complex structure $\mathcal{A}=\{\sigma_\alpha:\Delta^{n_\alpha}\rightarrow X\}_\alpha$ on a topological space $X$ and I have a quotient space $$\pi:X\rightarrow X'=X/\sim$$ Then we have a canonical family of maps $\mathcal{A}'=\{\pi \circ\sigma_\alpha:\Delta^{n_\alpha}\rightarrow X'\}_\alpha$ and hopefully this is going to be a $\Delta$-complex structure for $X'$. So perhaps this is the $\Delta$-complex structure that Hatcher gives.

But now if we take $X$ to be a square and we give it the $\Delta$-complex structure of the figure

Then this structure is explicitly given by the maps $$\mathcal{A}=\{\sigma_{ABD},\ \sigma_{BDC}, \ \sigma_{AB}, \sigma_{BD}, \ \sigma_{DA}, \ \sigma_{CB}, \ \sigma_{DC}, \ \sigma_{A}, \ \sigma_{B}, \ \sigma_{C}, \ \sigma_{D} \}$$

And then if we take the family of maps $\mathcal{A}'$ as above, this gives a $\Delta$-complex structure in the case of the torus. But it's not a $\Delta$-complex structure in the case of $\mathbb{R}P^2$ because in this case $\pi\circ \sigma_{AD} \neq \pi \circ \sigma_{BC}$ (because if $\Delta^1=[v_0,v_1]$ we have $\bar{A}=\pi\circ \sigma_{AB}(v_0)\neq \pi \circ \sigma_{DC}(v_0) = \bar{D}$) but $\text{im}(\pi\circ \sigma_{AD})=\text{im}(\pi\circ \sigma_{DC})$ so this mess up the partition part of the definition. I think there is a similar issue with the Klein bottle.

So my questions are

How you use the diagram above to find a $\Delta$-complex structure for the real projective plane and the Klein bottle and how do you do this for other quotients of $\Delta$-complexes (such as seeing the "dunce hat" as a quotient of a triangle).

Answer 1

The identification diagrams are not quotients of a delta complex, but rather delta complex structures on the quotient space for the square itself. Delta complexes don't behave particularly well under taking quotients, which is what I believe you are observing.

For example, with $T^2$ the identification diagram is representing the torus as $X=[0,1]\times[0,1]/{\sim}$ where $(0,t)\sim(1,t)$ and $(t,0)\sim(t,1)$ for all $t\in[0,1]$. The delta complex structure is

• $v:\Delta^0\to X$ by $v(1)=(0,0)$.
• $a:\Delta^1\to X$ by $a(1-t,t)=(0,t)$.
• $b:\Delta^1\to X$ by $b(1-t,t)=(t,0)$.
• $c:\Delta^1\to X$ by $c(1-t,t)=(t,t)$.
• $U:\Delta^2\to X$ by $U(t,1-t-s,s)=(s,1-t)$.
• $L:\Delta^2\to X$ by $V(t,1-t-s,s)=(1-t,s)$.

The images here should be understood as being in the quotient space. If I got the coordinates and orientations correct, then the boundaries of each of these simplices should be what the diagrams would lead you to expect.

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From another point of view, the identification diagrams are describing abstract delta complexes whose topological realizations are homeomorphic to the spaces Hatcher is claiming they are.

Answer 2

You say that

Then this structure is explicitly given by the maps $$\mathcal{A}=\{\sigma_{ABD},\ \sigma_{BDC}, \ \sigma_{AB}, \sigma_{BD}, \ \sigma_{DA}, \ \sigma_{CB}, \ \sigma_{DC}, \ \sigma_{A}, \ \sigma_{B}, \ \sigma_{C}, \ \sigma_{D} \}$$

but $\mathcal{A}$ is not a $\Delta$-complex structure on the square. For example, by definition of $\Delta$-complex, the restriction $\sigma_{ABD}\vert\ _{[e_0,e_2]}=\sigma_{AD}$ has to be in $\mathcal{A}$, but your collection does not contain it.

The main confusion arises from the fact that the $\Delta$-complex structure on a triangle is defined in such a way that the "orientations" of the edges are not in the cyclic order, but in the order which preserves the ordering of the vertices.