Let G = < A | R > be a presentation of a group.
Tietze Transformations (finite versions) are ways that a group presentation can be altered without changing the group.
\(T_I \) – add a relator that is a consequence of the other relators.
example: \(aba^{-1}b^{-1} , aba \) implies \(a^2b \)
\(T_{II} \) – add a new generator while simultaneously adding a relator that expresses that new generator as a word in the existing generators.
example: \(<a, b | aba^{-1}b^{-1}, aba> \) transformed into \(<a, b, c| c= a^2b^3, aba^{-1}b^{-1} , aba > \)
\(T_{I}^{-1}, T_{II}^{-1} \) – reverses of the above moves.
Fact: Finite presentation < A | R > and < B | S > represent isomorphic groups if one may be obtained from the other via a finite sequence of Tietze Transformations.
Note: Also true for non-finitely presented groups if we generalize the Tietze Transformation.
Goal: To construct a compact contractible 4 – manifold, that is not homeomorphic to a \(B^4 \).
We will construct a Mazur 4-manifold. We begin with a 4 -dimensional solid torus \(S^2 \times B^3 \).
We will attach a single 2-handle which kills the fundamental group but does not yield a 4-ball.
Clearly \(\partial (S^1 \times B^3 ) = S^1 \times S^2 \)
Begin by creating a picture of \(S^2 \times S^2 \) by doing a single 0 – Dehn surgery on the punctured unknot.
Next, we attach a 2 – handle whose attaching sphere (i.e. circle) lies in the complement of that unknot and hence in \(S^1 \times S^2 \).
That circle is pictured.
For meridional curve in the corresponding integral Dehn surgery, choose a simple closed curve that stays parallel to \(\Gamma \) in the given picture.
\(\pi_1 (S^3 – \Gamma \cup \Theta ) = < x_1 , … , x_9 | r_1 , … , r_9 >\)
\(\pi_1(S^1 \times S^2 – \Gamma ) = < x_1 , …, x_9 | r_1, … r_9 , x_5 x_2^{-1}x_1^{-1} > \)
\(\pi_1 ( \partial M^4 ) = <x_1 , … , x_9 | r_1, …, r_9, x_5x_2^{-1}x_1^{-1}, x_7^{-1} x_5^{-1} x_7 x_3^{-1} x_2^{-1} x_7^{-1} > \)
Perform Tietze Transformations
Let \(\beta = x_7 , \lambda = x_2 , \alpha = \beta \lambda \)
Perform Tietze Transformations
\(\pi_1 ( \partial M^4 ) \equiv < \alpha , \beta | \beta^5 = \alpha^7 , \beta^4 = \alpha^2 \beta \alpha^2 > \)
Goal: To show that this group is non-trivial.
First, we will introduce an additional relator \(\beta^5 = 1 \)
The resulting group is the quotient group \(\pi_1 (\partial M^4 ) / << \beta^5 >> \). If it is non-trivial then so is \(\pi_1 ( \partial M^4 ) \).
We now have \(< \alpha, \beta | \alpha^7 , \beta^5, \beta^4 = \alpha^2 \beta \alpha^2 > \)
