Calculating Homology Groups of RP(2)

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In summary, the conversation discusses difficulties with computing homology groups for simple spaces and suggests two approaches for calculating the homology groups of RP(2), which is a non-orientable manifold. The first approach involves using the fact that the universal covering space is the 2-sphere and the second approach involves representing RP(2) as a circle with a disc attached.
  • #1
Sephi
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I'm currently learning some homology theory but I have some difficulties computing homology groups of a few simple spaces. If someone could do the explicit calculation for RP(2), it would be really nice.

Thank you :)
 
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  • #2
the universal covering space is the 2 sphere, so pi 1 is Z/2Z, hence also the first homology group. that does it since the space is connected non orientable manifold so the zeroth homology is Z and the second homology i guess is zero.

i am just recalling this from 40 years ago since they don't let me teach topology for some reason, so i could be wrong.
 
  • #3
another approach is to represent RP^2 as a circle with a disc attached by a map of degree 2. then there is a little formula for the homology groups, in the chapter on cell complexes.
 

1. What is a homology group?

A homology group is a mathematical concept used in algebraic topology to study the properties of topological spaces. It is a collection of algebraic objects that describe the number of holes and higher dimensional structures in a space.

2. What is RP(2)?

RP(2) is the real projective plane, which is a two-dimensional space that can be thought of as the set of all lines passing through the origin in three-dimensional space. It is a non-orientable surface, meaning that it does not have a distinct front and back side.

3. How are homology groups of RP(2) calculated?

The homology groups of RP(2) are calculated using algebraic topology techniques, specifically the method of singular homology. This involves breaking down the space into simpler pieces and assigning algebraic objects to these pieces, which can then be manipulated to determine the homology groups.

4. What are the dimensions of the homology groups of RP(2)?

The homology groups of RP(2) have dimensions 1, 0, and 1 for the first, second, and third homology groups respectively. This means that RP(2) has one connected component, no holes, and one two-dimensional void or "cavity".

5. What is the significance of calculating homology groups of RP(2)?

Calculating the homology groups of RP(2) allows us to understand the topological structure of this space and how it differs from other surfaces. It also has applications in fields such as physics and computer science, where topological concepts are used to study complex systems and data.

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