The same question has been asked before, for instance here Hatcher exercise 2.1.2 deformation retract of $\Delta$-complex to Klein bottle by edge identifications and here Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle..
But I don't understand the answers there at all. This is perhaps due to the possibly different definitions that I'm using. The following is what I am trying to answer.
Show that the Δ complex obtained from $Δ^3$ by performing the order-preserving edge identifications $[v_0,v_1] ∼ [v_1,v_3]$ and $[v_0,v_2] ∼ [v_2,v_3]$ deformation retracts onto a Klein bottle. Also, find other pairs of identifications of edges that produce Δ complexes deformation retracting onto a torus, a 2 sphere, and $\mathbb RP^2$ .
Here are the definitions that I am using:
Semi simplicial sets: $X$, where $X$ consists of sets $X_n$ and maps called face maps $d_n:X_n\to X_{n-1}$, which have the property that $d_id_j=d_{j-1}d_i$ for $i<j$.
For a semi simplicial set $X$, I define the topological space $|X|$ as $|X|:=\frac{\sqcup X_n\times \Delta^n}{\sim}$, where $\Delta^n$ is the standard $n-$ simplex and $\sim$ is the equivalence relation generated by $(d_i(x), (x_0,x_1,...,x_{n-1}))\sim (x,(x_0,x_1,...,x_{i-1},0, x_i,...,x_n))$.
$\Delta$ complex: A space $Z$ such that $Z\cong |X|$ for some semi simplicial set $X$.
I don't understand how to even obtain the $\Delta$ complex in the question that I'm trying to answer using the above definitions.
Can anyone please
i) help me with this and also suggest me a reference where I can learn this?
ii) suggest me a reference which has lot of examples on algebraic topology for a beginner (the one who has studied continuity, compactness, connectedness etc. from Munkres and knows some algebra also but not so proficient with projective/flat modules, tensor products etc.) and also contains solutions? A reference like https://www.amazon.com/Problems-Real-Analysis-Advanced-Calculus/dp/0387773789/ref=sr_1_16?crid=2E19NIN0PIRX3&keywords=titu+andreescu&qid=1678603983&sprefix=Titu+and%2Caps%2C296&sr=8-16 but for algebraic topology will also be fine.
