Fundamental Group of Matrices

[curtdbz]

I am reading Munkres and know exactly how to find the fundamental groups of surfaces, using pi_1 and reducing it down to simpler problems. However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices! How do you go about doing that! There are no explanations in teh book, or the net (that I can find). The two main ones I'm having trouble with are here, if you could quickly help that would be great.

Find the fundamental groups of
1) The space of 2x2 matrices over C (complex) with 0 determinant with the topology given by the embedding in the vector space of all 2x2 matrices over C.
2) The space of upper triangular matrices of the size 2x2 over C with determinant 1 considered as a closed subspace in the vector space of all matrices size 2x2 over C.

I know the answer to 1, because someone told me it, but I don't know how they came to that conclusion. The answer to one is the integers, Z. Thank you!

 

[quasar987]

1 seems hard to me, but the space of question 2 looks contractible: write

a b
c d

for a 2x2 matrix with complex entries. Then the space of such matrices, is, as specified in the question, identified with C^4, and so the space of upper triangular matrices that concerns us is identified with the subspace of C^4 consisting of those quadruples (a,b,0,d) such that ad=1. That is, such that a =/= 0 and d=a^-1=(complex conjugate of a)/|a|². Surely, this space deformation retracts onto, say (1,1,0,1).

 

[wofsy]

Det(tA) = (t^n)Det(A) for an nxn matrix.

This gives a strong deformation retraction of the matrices of zero determinant onto the zero matrix.

(A,t) -> tA for t in the unit interval.

Thus the answer to 1 is the trivial group.

For 2 let the matrix a b o d go to the matrix a tb o d. This defines a strong deformation retraction onto the diagonal matrices of determinant 1. These matrices are homeomorphic to the non-zero complex numbers by the map z -> z 0 0 1/z.

The non-zero complex numbers deform onto the complex numbers of norm 1 which is the unit circle in the complex plane. Thus the fundamental group is the fundamental group of the circle which is Z.

The answer to 2 is the integers.

 

[berkeman]

However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices!

I know the answer to 1, because someone told me it, but I don't know how they came to that conclusion.

Sounds like you are asking us to do your final take-home exam for you...?

 

[blue_raver22]

The fundamental group of a topological space is a mathematical concept that captures the idea of "holes" or "loops" in the space. In the case of surfaces, this can be visualized as loops that can be continuously deformed without leaving the surface. However, when we consider spaces of matrices, the concept of "holes" or "loops" may not be as intuitive.

To find the fundamental group of a space of matrices, we need to first understand the topology of the space. In the first problem, we are considering the space of 2x2 matrices over C with 0 determinant. This space can be embedded in the vector space of all 2x2 matrices over C, which has a natural topology induced by the norm on matrices. This means that the topology of our space of 2x2 matrices will also be induced by the norm.

In general, the fundamental group of a topological space can be defined as the set of all homotopy classes of loops in the space, with a binary operation of concatenation. A homotopy is a continuous deformation of one loop into another, and two loops are considered in the same homotopy class if they can be continuously deformed into each other without leaving the space.

In the case of our space of 2x2 matrices, we can think of a loop as a continuous path starting at the identity matrix and ending at a matrix with 0 determinant. This is because a matrix with 0 determinant cannot be continuously deformed into the identity matrix without leaving the space. In this case, the fundamental group will consist of all homotopy classes of such loops, and the binary operation of concatenation will be matrix multiplication.

Now, to find the fundamental group, we need to determine the homotopy classes of loops in our space. It turns out that any loop in this space can be continuously deformed into a loop that consists of a sequence of diagonal matrices with determinant 1, followed by a single matrix with 0 determinant. This is because any matrix with 0 determinant can be written as a product of diagonal matrices with determinant 1. This means that the fundamental group of this space is isomorphic to the fundamental group of the space of diagonal matrices with determinant 1, which is isomorphic to the integers (Z).

For the second problem, we are considering the space of upper triangular matrices of size 2x2 over C with determinant 1, which is a closed