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X1 = {(x; y; z) ∈ R^3 | x > 0}
just need to check my thinking
is pi1(X1) = {1} i.e. trivial
just need to check my thinking
is pi1(X1) = {1} i.e. trivial
Mikemaths said:I have done some more research and now I understand that the fundamental group of X is the group consisting of the homotopic equivalence classes of loops of base x in X. But this must be expressed in algebraic terms, now I understand that {2} is not a group but I beleiev there are two different equivalence classes of loops in X2 as described above as the loops either side of the x-axis cannot be equivalent?
Am i talking rubbish or is this valid
Well exactly! Any loop in R^3\{(x; y; z) | x = 0; y = 0; 0 <= z <= 1} can be homotoped to the constant loop by avoiding the unit line on the z axis just as any loop in R^3\{(x; y; z) | x = 0; y = 0; z = 0} can be homotoped to the constant loop by avoiding the origin. So by this argument, both spaces have trivial fundamental group. (And it makes sense to speak of the fundamental group of these spaces without reference to a particular base point because they are path connected spaces, so the fundamental group is independant of the base point (up to isomorphy of course).)Mikemaths said:I am not sure since a loop from x to x in
R^3\{(x; y; z) | x = 0; y = 0; 0 <= z <= 1}
is a similar situation to
R^3\{(x; y; z) | x = 0; y = 0; z = 0}
As they can always avoid the unit line on z axis that is missing as it were.
This space is not isomorphic (we say homeomorphic) to the torus, why would you think that?Mikemaths said:Also is R^3\{(x; y; z) | x = 0; 0 <= y <= 1}
isomorphic to the Torus and therefore fundamental group of this is Z + Z (disjoint)?
A fundamental group is a mathematical concept used in topology to study the properties of a topological space. It is a group that consists of all the possible loops in the space, where the group operation is the concatenation of loops.
A fundamental group can be calculated using algebraic topology techniques, such as the Van Kampen theorem or the Seifert-van Kampen theorem. These theorems provide a way to decompose a space into smaller, simpler spaces and then combine the fundamental groups of these spaces to obtain the fundamental group of the original space.
The fundamental group of a space is a topological invariant, which means it does not change under homeomorphisms. This allows us to distinguish between different topological spaces and classify them into different types. The fundamental group also helps to study the properties of a space, such as connectedness, compactness, and orientability.
Yes, the fundamental group of a space can be infinite. In fact, there are many examples of spaces with infinite fundamental groups, such as the circle and the torus. The size of the fundamental group depends on the complexity of the space and its topology.
The fundamental group is closely related to other mathematical concepts, such as homotopy, homology, and cohomology. These concepts help to study the properties of topological spaces and provide different ways to calculate the fundamental group. The fundamental group is also related to the fundamental groupoid, which is a generalization of the fundamental group for more complicated spaces.