Spin bordism
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[edit] 1 Introduction
By the Pontrjagin-Thom isomorphism the spin bordism groups of closed manifolds with spin structures are isomorphic to the homotopy groups of the Thom spectrum .
Preliminary results were by Novikov. The main calculation was achieved in [Anderson&Brown&Peterson1966] and [Anderson&Brown&Peterson1967].
[edit] 2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.
For a multi-index , we set and .
A spin structure on a closed -manifold induces a KO-orientation , so that we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbersThere is an interpretation of these characteristic numbers using index theory:
A feature of Spin manifolds is that they possess Dirac operators, . The (Clifford-linear) Dirac operator can be considered as a representative of the fundamental class , see [Atiyah1970] and [Higson&Roe2000]. The characteristic numbers above can then be defined as the indices of the (Clifford-linear) Dirac operators obtained by twisting with the corresponding vector bundles. The easiest case is the non-twisted one: is the trivial bundle, and taking the index of the Dirac operator defines an element of when is n-dimensional. This gives rises to a ring homomorphism often called the -invariant:
See [Lawson&Michelsohn1989]. For divisible by 4, this invariant equals (up to a factor of 1/2 in dimensions congruent to 4 modulo 8) the -genus of .
[edit] 3 Classification
[edit] 3.1 MSpin away from the prime 2 and at the prime 2
After inverting 2 the map of Thom spectra becomes a homotopy equivalence.
Using the Thom isomorphism we get for each multi-index with a map , which induces on homotopy groups the map , described above. Anderson,Brown and Peterson show that factorizes through if is even and if is odd. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.
Similarly a Stiefel-Whitney class in corresponds to a spectrum map .
Theorem 3.1 [Anderson&Brown&Peterson1967]. There are classes such that there is a 2-local homotopy equivalence
[Anderson&Brown&Peterson1966] also determine the Poincaré polynomial of which allows to compute inductively the degrees . The first occurs in dimension , and there are in all dimensions .
[edit] 3.2 Consequences
From this one can compute the additive structure of the spin bordism groups completely. We get a contribution from each which is 0 below dimension , and periodic of period 8 starting from dimension , with values (here the first value corresponds to dimensions congruent to 0 modulo 8). The contribution from is a single in dimension .
All summands are in degrees divisible by 4, and there is no odd torsion in the spin cobordism groups. All even torsion is of exponent 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism.
[edit] 3.3 Ring structure
Concerning the multiplicative structure, is the subring of an integral polynomial ring on classes (dimension 4i) consisting of all classes of dimension a multiple of 8 and twice the classes whose dimension is not a multiple of 8 [Stong1968]. Anderson, Brown and Peterson determine the structure of modulo the ideal consisting of torsion mapping monomorphically into unoriented cobordism. According to [Laures2003], the multiplicative structure of this ideal is still not completely known.
[edit] 4 Generators
[edit] 4.1 Low dimensions
The spin bordism groups up to dimension are given in [Milnor1963a] without proof. Milnor states that this is the result of a formibable calculation of for .
, generated by a point.
, generated by , the circle with the "antiperiodic" spin structure.
, generated by .
.
, generated by , the Kummer surface.
.
, generated by quaternionic projective space and a generator which equals 1/4 of the square of the Kummer surface.
[edit] 4.2 Generators in all dimensions as given by the classification
By the theorem of Anderson, Brown and Peterson there exist manifolds of dimension if is even, of dimension if is odd, and of dimension , such that the characteristic numbers , and are odd. For odd, let be a spin nullbordism of , and let (using an orientation-reversing automorphism of ). By [Stong1966] there exist manifolds such that . If all are even, one can choose to be a product of quaternionic projective spaces.
Then a basis for is given by
- for
- for .
A basis for is given by
- with , and even,
- with , even,
- with , odd,
- with , and odd
- with odd,
- .
[edit] 5 Further topics
[edit] 5.1 Rohlin's theorem
Above we stated that the 4-dimensional spin bordism group is generated by the Kummer surface which has signature 16. Consequently we have the following important theorem of Rohlin (which of course was used in calculation of give above).
Theorem 5.1 [Rohlin1952]. The signature of every closed smooth spin -manifold is divisible by .
[edit] 5.2 Relationship with other bordism groups
- Framed bordism: the image of is 0 unless or when it is and detected by the -invariant. Explicit generators are with and .
- Oriented bordism: the kernel of lies in dimensions and . It is a vector space with a basis , for even, and , for odd, where and . It is also the ideal generated by the non-trivial class of . The cokernel is a finite -torsion group which is trivial if and only if or equivalently , , , , and . [Milnor1965] computed .
- Unoriented bordism: the image of is all bordism classes for which the characteristic numbers divisible by and are zero. A basis for the image consists of the , for even, the , for odd and the . The image is trivial for . In even dimensions it is additionally trivial for and . In odd dimensions it is trivial for and also for and . Otherwise the image is non trivial.
[edit] 6 References
- [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605
- [Anderson&Brown&Peterson1967] D. W. Anderson, E. H. Brown and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR0219077 (36 #2160) Zbl 0156.21605
- [Atiyah1970] M. F. Atiyah, Global theory of elliptic operators, (1970), 21–30. MR0266247 (42 #1154) Zbl 0193.43601
- [Higson&Roe2000] N. Higson and J. Roe, Analytic -homology, Oxford University Press, Oxford, 2000. MR1817560 (2002c:58036) Zbl 1146.19004
- [Laures2003] G. Laures, An splitting of spin bordism, Amer. J. Math. 125 (2003), no.5, 977–1027. MR2004426 (2004g:55007) Zbl 1058.55001
- [Lawson&Michelsohn1989] H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) Zbl 0801.58017
- [Milnor1963a] J. Milnor, Spin structures on manifolds, Enseignement Math. (2) 9 (1963), 198–203. MR0157388 (28 #622) Zbl 0116.40403
- [Milnor1965] J. W. Milnor, Remarks concerning spin manifolds, in Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, (1965) 55–62. MR0180978 (31 #5208) Zbl 0132.19602
- [Rohlin1952] V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221–224. MR0052101 (14,573b)
- [Stong1966] R. E. Stong, Relations among characteristic numbers. II, Topology 5 (1966), 133–148. MR0192516 (33 #741) Zbl 0142.40902
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
Preliminary results were by Novikov. The main calculation was achieved in [Anderson&Brown&Peterson1966] and [Anderson&Brown&Peterson1967].
[edit] 2 Invariants
The spin bordism class of a manifold is detected by -cohomology (Stiefel-Whitney) and KO-theory (Pontryagin) characteristic numbers.
For a multi-index , we set and .
A spin structure on a closed -manifold induces a KO-orientation , so that we can evaluate polynomials in the KO-Pontryagin classes to get characteristic numbersThere is an interpretation of these characteristic numbers using index theory:
A feature of Spin manifolds is that they possess Dirac operators, . The (Clifford-linear) Dirac operator can be considered as a representative of the fundamental class , see [Atiyah1970] and [Higson&Roe2000]. The characteristic numbers above can then be defined as the indices of the (Clifford-linear) Dirac operators obtained by twisting with the corresponding vector bundles. The easiest case is the non-twisted one: is the trivial bundle, and taking the index of the Dirac operator defines an element of when is n-dimensional. This gives rises to a ring homomorphism often called the -invariant:
See [Lawson&Michelsohn1989]. For divisible by 4, this invariant equals (up to a factor of 1/2 in dimensions congruent to 4 modulo 8) the -genus of .
[edit] 3 Classification
[edit] 3.1 MSpin away from the prime 2 and at the prime 2
After inverting 2 the map of Thom spectra becomes a homotopy equivalence.
Using the Thom isomorphism we get for each multi-index with a map , which induces on homotopy groups the map , described above. Anderson,Brown and Peterson show that factorizes through if is even and if is odd. Hence the corresponding characteristic numbers vanish for manifolds of smaller dimension.
Similarly a Stiefel-Whitney class in corresponds to a spectrum map .
Theorem 3.1 [Anderson&Brown&Peterson1967]. There are classes such that there is a 2-local homotopy equivalence
[Anderson&Brown&Peterson1966] also determine the Poincaré polynomial of which allows to compute inductively the degrees . The first occurs in dimension , and there are in all dimensions .
[edit] 3.2 Consequences
From this one can compute the additive structure of the spin bordism groups completely. We get a contribution from each which is 0 below dimension , and periodic of period 8 starting from dimension , with values (here the first value corresponds to dimensions congruent to 0 modulo 8). The contribution from is a single in dimension .
All summands are in degrees divisible by 4, and there is no odd torsion in the spin cobordism groups. All even torsion is of exponent 2, being of two types: that arising by products with a framed and that which maps monomorphically into unoriented cobordism.
[edit] 3.3 Ring structure
Concerning the multiplicative structure, is the subring of an integral polynomial ring on classes (dimension 4i) consisting of all classes of dimension a multiple of 8 and twice the classes whose dimension is not a multiple of 8 [Stong1968]. Anderson, Brown and Peterson determine the structure of modulo the ideal consisting of torsion mapping monomorphically into unoriented cobordism. According to [Laures2003], the multiplicative structure of this ideal is still not completely known.
[edit] 4 Generators
[edit] 4.1 Low dimensions
The spin bordism groups up to dimension are given in [Milnor1963a] without proof. Milnor states that this is the result of a formibable calculation of for .
, generated by a point.
, generated by , the circle with the "antiperiodic" spin structure.
, generated by .
.
, generated by , the Kummer surface.
.
, generated by quaternionic projective space and a generator which equals 1/4 of the square of the Kummer surface.
[edit] 4.2 Generators in all dimensions as given by the classification
By the theorem of Anderson, Brown and Peterson there exist manifolds of dimension if is even, of dimension if is odd, and of dimension , such that the characteristic numbers , and are odd. For odd, let be a spin nullbordism of , and let (using an orientation-reversing automorphism of ). By [Stong1966] there exist manifolds such that . If all are even, one can choose to be a product of quaternionic projective spaces.
Then a basis for is given by
- for
- for .
A basis for is given by
- with , and even,
- with , even,
- with , odd,
- with , and odd
- with odd,
- .
[edit] 5 Further topics
[edit] 5.1 Rohlin's theorem
Above we stated that the 4-dimensional spin bordism group is generated by the Kummer surface which has signature 16. Consequently we have the following important theorem of Rohlin (which of course was used in calculation of give above).
Theorem 5.1 [Rohlin1952]. The signature of every closed smooth spin -manifold is divisible by .
[edit] 5.2 Relationship with other bordism groups
- Framed bordism: the image of is 0 unless or when it is and detected by the -invariant. Explicit generators are with and .
- Oriented bordism: the kernel of lies in dimensions and . It is a vector space with a basis , for even, and , for odd, where and . It is also the ideal generated by the non-trivial class of . The cokernel is a finite -torsion group which is trivial if and only if or equivalently , , , , and . [Milnor1965] computed .
- Unoriented bordism: the image of is all bordism classes for which the characteristic numbers divisible by and are zero. A basis for the image consists of the , for even, the , for odd and the . The image is trivial for . In even dimensions it is additionally trivial for and . In odd dimensions it is trivial for and also for and . Otherwise the image is non trivial.
[edit] 6 References
- [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605
- [Anderson&Brown&Peterson1967] D. W. Anderson, E. H. Brown and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR0219077 (36 #2160) Zbl 0156.21605
- [Atiyah1970] M. F. Atiyah, Global theory of elliptic operators, (1970), 21–30. MR0266247 (42 #1154) Zbl 0193.43601
- [Higson&Roe2000] N. Higson and J. Roe, Analytic -homology, Oxford University Press, Oxford, 2000. MR1817560 (2002c:58036) Zbl 1146.19004
- [Laures2003] G. Laures, An splitting of spin bordism, Amer. J. Math. 125 (2003), no.5, 977–1027. MR2004426 (2004g:55007) Zbl 1058.55001
- [Lawson&Michelsohn1989] H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) Zbl 0801.58017
- [Milnor1963a] J. Milnor, Spin structures on manifolds, Enseignement Math. (2) 9 (1963), 198–203. MR0157388 (28 #622) Zbl 0116.40403
- [Milnor1965] J. W. Milnor, Remarks concerning spin manifolds, in Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, (1965) 55–62. MR0180978 (31 #5208) Zbl 0132.19602
- [Rohlin1952] V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221–224. MR0052101 (14,573b)
- [Stong1966] R. E. Stong, Relations among characteristic numbers. II, Topology 5 (1966), 133–148. MR0192516 (33 #741) Zbl 0142.40902
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010