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To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots on nice pictures!) (Edit: I mean for quite general presentations.)

To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots of nice pictures!) (Edit: I mean for quite general presentations.)

To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots on nice pictures!)

To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots on nice pictures!) (Edit: I mean for quite general presentations.)

To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000. Book on AMS site) In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots on nice pictures!)

To add to Terry Tao's excellent description of some of the geometric ideas behind the Hall-Witt identity, I would draw attention to the interpretation given by Loday in his paper on Homotopical Syzygies (in Une dégustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, AMS, 2000; book on AMS site). In his section 1.4. in examining the syzygies / identities amongst the relations of the obvious presentation of the free Abelian group on three symbols, he shows how the commutators/ 2-syzygies are drawn as the faces of the Cayley graph considered as being embedded in a sphere (and thus appearing as an empty cube). There is a 3-syzygy needed to build the next stage of a resolution of the group and that is a 3-cell filling the cube. The boundary of that 3-cell can be read off as the Jacobi-Witt-Hall (JWH) identity (with slightly different conventions more as in Terry's article than in the original form in the question).

Thus the proof of freeness of the commutator subgroup of $F(x,y,z)$ does not give a useful basis for that subgroup and the commutators and their conjugates are not without relations between them. The JWH identity is one of those relations. The exploration of homotopical syzygies throws up some lovely calculations in general (and lots on nice pictures!)