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

algebraic-topologyhomological-algebrasimplicial-complex

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

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

enter image description here

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).

Best Answer

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.

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.

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