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I have to learn how to solve problems like the following in the next two weeks:

Let $X$ be the quotient space obtained from an 8-sided polygonal region P by pasting its edges together according to the labelling scheme $aabbcdc^{-1}d^{-1}$.

The problem is, I don't know where to start. What does this even mean? I know only some very basic things about algebraic topology, e.g. the fundamental group, covering maps, deformation retracts, and homotopy equivalences.

D_S
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  • If you have to do something, and you don't know the terms, perhaps you either have a book or a professor or a TA? – Thomas Andrews Dec 23 '14 at 04:10
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    This appears to be related to a problem in Munkres' book. http://math.stackexchange.com/questions/37696/quotient-space-from-the-8-sided-polygonal-region-munkres – Thomas Andrews Dec 23 '14 at 04:11
  • Write along the edges each of the labels and an arrow in the clockwise direction for $a,b,c,d$ and counter-clockwise for $a^{-1},b^{-1},c^{-1},d^{-1}$. Then identify the points on the two segments labeled $a$ according to the arrows, etc. – Thomas Andrews Dec 23 '14 at 04:15
  • There are many good books on topology which explain in depth the topology of surfaces, and in particular explain how to study surfaces by gluing polygons together according to this kind of labelling scheme. I recommend "Classical topology and Combinatorial group theory" by John Stillwell. – Lee Mosher Dec 23 '14 at 20:13

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This section of the Wikipedia article on surfaces gives a good overview of what this means.

Basically, if you see $s$, you label the next edge of the polygon with a clockwise arrow and the label $s$, and where you see $s^{-1}$ you label it $s$ with a counter-clockwise arrow.

Then you identify the points of two edges with the same symbol so the arrows are in the same direction.

Thomas Andrews
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A precise answer assumes that you have a good understanding of quotient spaces and their relation with equivalence classes.

Assume that each edge of the polygon is assigned a parameterization by the domain $[0,1]$ which goes in the counterclockwise direction around the boundary of the polygon. For instance, if your polygon is literally a Euclidean polygon with straight sides then a side $PQ$ with initial vertex $P$ and terminal vertex $Q$ can be parameterized (using vector operation) as $$\gamma(t) = P + t \, (Q-P) $$ Define the "gluing relation" to be the smallest equivalence relation $\sim$ on the polygon satisfying the following:

  • If two sides of the polygon have parameterizations $\gamma_1,\gamma_2$, and if those two sides are labelled with the same letter and the same sign by the given labeling scheme, then $\gamma_1(t)\sim\gamma_2(t)$.

  • If two sides of the polygon have parameterizations $\gamma_1,\gamma_2$, and if those two sides are labelled with the same letter and opposite signs by the given labelling scheme, then $\gamma_1(t) \sim \gamma_2(1-t)$.

Then form the quotient space of this equivalence relation.

One can prove that the quotient space is well-defined up to homeomorphism independent of the choices of parameterizations, depending only on the "labelling scheme".

Lee Mosher
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