Reidemeister Twist

In the mathematical sub-area of topology , the Reidemeister torsion (also Reidemeister-Franz torsion ) is a topological invariant that can also be used to distinguish spaces for which classic invariants of algebraic topology such as fundamental groups and homology groups match.

A variant of today's construction was used by Kurt Reidemeister in 1935 to classify the homeomorphism types of 3-dimensional lens spaces . A little later, Wolfgang Franz used the construction shown below in order to be able to classify higher-dimensional lens spaces.

construction

Let be a compact CW complex with vanishing Euler characteristics and a representation of the fundamental group . ${\displaystyle K}$ ${\displaystyle \chi (K)=0}$ ${\displaystyle \rho \colon \pi _{1}K\to SL(n,\mathbb {C} )}$

Be the universal overlay that acts on through deck movements and its singular chain complex . The twisted chain complex ${\displaystyle {\widetilde {K}}}$ ${\displaystyle \pi _{1}K}$ ${\displaystyle C_{*}({\widetilde {K}},\mathbb {Z} )}$

${\displaystyle C_{*}(K,\rho ):=C_{*}({\widetilde {K}},\mathbb {Z} )\otimes _{\rho }\mathbb {C} ^{n}}$

is the quotient of the tensor product under the identification for all . The edge operator of induces an edge operator on . In order to be able to define the Reidemeister torsion, we have to assume that is acyclic, that is . ${\displaystyle C_{*}({\widetilde {K}},\mathbb {Z} )\otimes _{\mathbb {Z} }\mathbb {C} ^{n}}$ ${\displaystyle v\otimes \gamma ^{-1}c=\rho (\gamma )v\otimes c}$ ${\displaystyle \gamma \in \pi _{1}K}$ ${\displaystyle \partial _{\overline {K}}}$ ${\displaystyle C_{*}({\widetilde {K}},\mathbb {Z} )}$ ${\displaystyle \partial :=\partial _{\overline {K}}\otimes id}$ ${\displaystyle C_{*}(K,\rho )}$ ${\displaystyle C_{*}(K,\rho )}$ ${\displaystyle ker(\partial )=im(\partial )}$

Now be the subgroup of the margins . Choose a base of and continue with the base complement sentence to a base of . Because of that we have an exact sequence ${\displaystyle B_{i}:=Bild(\partial )\subset C_{i}(K,\rho )}$ ${\displaystyle \left\{b_{k}^{i}\right\}_{k}}$ ${\displaystyle B_{i}}$ ${\displaystyle \left\{c_{k}^{i}\right\}_{k}}$ ${\displaystyle C_{i}(K,\rho )}$ ${\displaystyle H_{i}(K,\rho )=0}$

${\displaystyle 0\to B_{i}\to C_{i}(K,\rho )\to B_{i-1}\to 0}$

and can find the archetypes , so that is a basis of . ${\displaystyle b_{l}^{i-1}\in B_{i-1}}$ ${\displaystyle {\tilde {b}}_{l}^{i-1}\in C_{i}(K,\rho )}$ ${\displaystyle \left\{b_{k}^{i}\right\}_{k}\cup \left\{{\tilde {b}}_{l}^{i-1}\right\}_{l}}$ ${\displaystyle C_{i}(K,\rho )}$

For the bases and there is a unique matrix that maps the first base to the second. We denote the determinant of this matrix with . Then we define the Reidemeister twist of through ${\displaystyle \left\{c_{k}^{i}\right\}_{k}}$ ${\displaystyle \left\{b_{k}^{i}\right\}_{k}\cup \left\{{\tilde {b}}_{l}^{i-1}\right\}_{l}}$ ${\displaystyle \left[b_{i}\cup {\tilde {b}}_{i-1}\colon c_{i}\right]}$ ${\displaystyle (K,\rho )}$

${\displaystyle \tau (K,\rho )=\pm \Pi _{i\geq 0}{\frac {\left[b_{2i}\cup {\tilde {b}}_{2i-1}\colon c_{2i}\right]}{\left[b_{2i+1}\cup {\tilde {b}}_{2i}\colon c_{2i+1}\right]}}\in \mathbb {C} ^{*}/\left\{\pm 1\right\}}$.

The uncertainty arises from the dependence of the determinant on the arrangement of the basic elements. All other choices have no influence on the result, in particular due to the alternating product the factors arising from the choice of another basis cancel each other out. ${\displaystyle \pm 1}$ ${\displaystyle b_{i}}$

Invariance

Reidemeister torsion is generally not invariant among homotopy equivalences and can therefore be used to distinguish homotopy-equivalent but not homeomorphic spaces. The Reidemeister torsion (for a given representation of the fundamental group) is invariant under simple homotopy equivalences .

A relative version of the Reidemeister torsion can be used to distinguish PL complexes that are homeomorphic but not PL-equivalent.

Examples

• For the lens space and the representation with for a -th root of unit one obtains , where the solution of , that is, the inverse of in denotes. In particular, one obtains for different Reidemeister torsions, which means that these lens spaces cannot be homeomorphic. ${\displaystyle L(p,q)}$ ${\displaystyle \rho _{\xi }\colon \pi _{1}M=\mathbb {Z} /p\mathbb {Z} \to SU(2)}$ ${\displaystyle \rho _{\xi }(1)=diag(\xi ,\xi ^{-1})}$ ${\displaystyle p}$ ${\displaystyle \xi }$ ${\displaystyle \tau (L(p,q),\rho _{\xi })=\mid \xi -1\mid \mid \xi ^{q^{-1}}-1\mid }$ ${\displaystyle q^{-1}}$ ${\displaystyle aq\equiv 1\ mod\ p}$ ${\displaystyle q}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle q^{\prime }\not \equiv \pm q^{\pm 1}\ mod\ p}$
• A spherical spatial shape is clearly defined by its fundamental group and its Reidemeister torsions in all representations . ${\displaystyle \pi _{1}M\to O(n)}$
• For one -dimensional rational homology sphere and the trivial representation is that Reidemeister torsion therefore related to the twist of the homology groups together. ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle \rho }$ ${\displaystyle \tau (M,\rho )=\Pi _{i=1}^{n-1}\mid H_{i}(M,\mathbb {Z} )\mid ^{(-1)^{i}}}$
• For a knot complement and the means of Abelisierung by given representation is the Alexander polynomial . ${\displaystyle ab\colon \pi _{1}(S^{3}\setminus K)\to H_{1}(S^{3}\setminus K)=\mathbb {Z} }$ ${\displaystyle \gamma \to t^{ab(\gamma )}}$ ${\displaystyle \rho \colon \pi _{1}(S^{3}\setminus K)\to \mathbb {Q} (t)}$ ${\displaystyle {\tfrac {t-1}{\tau (S^{3}\setminus K,\rho )}}}$

Cheeger-Müller's theorem

Cheeger-Müller's theorem states the equality of analytical torsion and Reidemeister torsion (except for the sign, because the Reidemeister torsion is only defined up to the sign). It was first proven by Cheeger and Müller for orthogonal or unitary representations and later generalized by Müller to unimodular representations.

literature

• John Milnor: Whitehead Torsion. Bull. Amer. Math. Soc. 72: 358-426 (1966).
• G. de Rham, S. Maumary, M. Kervaire: Torsion et type simple d'homotopie. Exposés faits au séminaire de topologie de l'Université de Lausanne. Lecture Notes in Mathematics, No. 48, Springer-Verlag, Berlin-New York, 1967.
• Vladimir Turaev: Torsions of 3-dimensional manifolds. Progress in Mathematics, 208. Birkhäuser Verlag, Basel, 2002. ISBN 3-7643-6911-6
• Kiyoshi Igusa: Higher Franz-Reidemeister torsion. AMS / IP Studies in Advanced Mathematics, 31. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. ISBN 0-8218-3170-4
• Liviu Nicolaescu: The Reidemeister torsion of 3-manifolds. De Gruyter Studies in Mathematics, 30. Walter de Gruyter & Co., Berlin, 2003. ISBN 3-11-017383-2

Web links

• Reidemeister Torsion (MathWorld)

Individual evidence

1. ^ TA Chapman: Topological invariance of Whitehead torsion. Amer. J. Math. 96: 488-497 (1974).
2. ^ John Milnor: Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2) 74: 575-590 (1961).
3. ^ Kurt Reidemeister: Homotopy rings and lens rooms . Dep. Math. Sem. Univ. Hamburg 11: 102-109 (1935).
4. ↑ Wolfgang Franz: About the torsion of an overlap. J. Reine Angew. Math. 173: 245-254 (1935).
5. ↑ Georges de Rham: Complexes à automorphismes et homéomorphie différentiable. Ann. Inst. Fourier Grenoble 2 (1950), 51-67 (1951).
6. ^ John Milnor: A duality theorem for Reidemeister torsion. Ann. of Math. (2) 76: 137-147 (1962).
7. ↑ Werner Müller: Analytic torsion and R-torsion of Riemannian manifolds. Adv. In Math. 28 (1978), no. 3, 233-305.
8. ^ Jeff Cheeger: Analytic torsion and the heat equation. Ann. of Math. (2) 109 (1979) no. 2, 259-322.
9. ↑ Werner Müller: Analytic torsion and R -torsion for unimodular representations. J. Amer. Math. Soc. 6, no. 3, 721-753 (1993).