Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Involutions on homotopy spheres and their gluing diffeomorphisms


Author: Chao Chu Liang
Journal: Trans. Amer. Math. Soc. 215 (1976), 363-391
MSC: Primary 57D65; Secondary 57E25
DOI: https://doi.org/10.1090/S0002-9947-1976-0431213-1
Erratum: Trans. Amer. Math. Soc. 222 (1976), 405.
MathSciNet review: 0431213
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ hS({P^{2n + 1}})$ denote the set of equivalence classes of smooth fixed-point free involutions on $ (2n + 1)$-dimensional homotopy spheres. Browder and Livesay defined an invariant $ \sigma ({\Sigma ^{2n + 1}},T)$ for each $ ({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})$, where $ \sigma \in Z$ if n is odd, $ \sigma \in {Z_2}$ if n is even. They showed that for $ n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0$ if and only if $ ({\Sigma ^{2n + 1}},T)$ admits a codim 1 invariant sphere. For any $ ({\Sigma ^{2n + 1}},T)$, there exists an A-equivariant diffeomorphism f of $ {S^n} \times {S^n}$ such that $ ({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)$, where A denotes the antipodal map. Let $ \beta (f) = \sigma ({\Sigma ^{2n + 1}},T)$. In the case n is odd, we can show that the Browder-Livesay invariant is additive: $ \beta (fg) = \beta (f) + \beta (g)$. But if n is even, then there exists f and g such that $ \beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)$. Let $ {D_0}({S^n} \times {S^n},A)$ be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of $ {S^n} \times {S^n}$. We can prove that ``For $ n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})$ is in 1-1 correspondence with a subgroup of $ {D_0}({S^n} \times {S^n},A)$. As an application of these theorems, we demonstrated that ``Let $ \Sigma _0^{8k + 3}$ denote the generator of $ b{P_{8k + 4}}$. Then the number of $ (\Sigma _0^{8k + 3},T)$'s with $ \sigma (\Sigma _0^{8k + 3},T) = 0$ is either 0 or equal to the number of $ ({S^{8k + 3}},T)$'s with $ \sigma ({S^{8k + 3}},T) = 0$, where $ {S^{8k + 3}}$ denotes the standard sphere".


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, On the groups $ J(X)$. II, Topology 3 (1965), 137-171. MR 33 #6626. MR 0198468 (33:6626)
  • [2] I. Berstein, Involutions with nonzero Arf invariant, Bull. Amer. Math. Soc. 74 (1968), 678-682. MR 38 #5225. MR 0236932 (38:5225)
  • [3] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [4] W. Browder, Structures on $ M \times R$, Proc. Cambridge Philos. Soc. 61 (1965), 337-345. MR 30 #5321. MR 0175136 (30:5321)
  • [5] -, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, Berlin and New York, 1972. MR 0358813 (50:11272)
  • [6] -, Cobordism invariants, the Kervaire invariant and fixed point free involutions, Trans. Amer Math. Soc. 178 (1973), 193-225. MR 48 #3067. MR 0324717 (48:3067)
  • [7] W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres, Bull. Amer. Math. Soc. 73 (1967), 242-245. MR 34 #6781. MR 0206965 (34:6781)
  • [8] -, Fixed point free involutions on homotopy spheres, Tôhoku Math. J. (2) 25 (1973), 69-87. MR 47 #9610. MR 0321077 (47:9610)
  • [9] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33, Springer-Verlag, Berlin; Academic Press, New York, 1964. MR 31 #750. MR 0176478 (31:750)
  • [10] A. Dold, Über fasernweise homotopieäquivalenz von Faserräumen, Math. Z. 62 (1955), 111-136. MR 17, 519. MR 0073986 (17:519d)
  • [11] M. Fujii, KO-groups of projective spaces, Osaka J. Math. 4 (1967), 141-149. MR 36 #2143. MR 0219060 (36:2143)
  • [12] A. Haefliger, Plongements différentiable de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47-82. MR 26 #3069. MR 0145538 (26:3069)
  • [13] M. W. Hirsch and J. Milnor, Some curious involutions of spheres, Bull. Amer. Math. Soc. 70 (1964), 372-377. MR 31 #751. MR 0176479 (31:751)
  • [14] D. Husemoller, Fibre bundles, McGraw-Hill, New York, 1966. MR 37 #4821. MR 0229247 (37:4821)
  • [15] I. James and E. Thomas, An approach to the enumeration problem for non-stable vector bundles, J. Math. Mech. 14 (1965), 485-506. MR 30 #5319. MR 0175134 (30:5319)
  • [16] M. A. Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161-169. MR 22 #4075. MR 0113237 (22:4075)
  • [17] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504-537. MR 26 #5584. MR 0148075 (26:5584)
  • [18] A. Kosinski, On the inertia group of $ \pi $-manifolds, Amer. J. Math. 89 (1967), 227-248. MR 35 #4936. MR 0214085 (35:4936)
  • [19] J. Levine, Self-equivalences of $ {S^n} \times {S^k}$, Trans. Amer. Math. Soc. 143 (1969), 523-543. MR 40 #2098. MR 0248848 (40:2098)
  • [20] G. R. Livesay and C. B. Thomas, Free $ {Z_2}$ and $ {Z_3}$ actions on homotopy spheres, Topology 7 (1968), 11-14. MR 36 #3343. MR 0220277 (36:3343)
  • [21] S. López de Medrano, Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 59, Springer-Verlag, New York, 1971. MR 45 #7747. MR 0298698 (45:7747)
  • [22] J. W. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962-972. MR 22 #990. MR 0110107 (22:990)
  • [23] R. E. Mosher and M. C. Tangora, Cohomology operations and application in homotopy theory, Harper & Row, New York and London, 1968. MR 37 #2223. MR 0226634 (37:2223)
  • [24] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458-480. MR 15, 338. MR 0058212 (15:338a)
  • [25] -, Cocycle formulas for homotopy classification; maps into projective and lens space, Trans. Amer. Math. Soc. 103 (1962), 30-44. MR 25 #576. MR 0137120 (25:576)
  • [26] P. Orlik, On the Arf invariant of an involution, Canad. J. Math. 22 (1970), 519-524. MR 41 #7718. MR 0263113 (41:7718)
  • [27] D. Puppe, Homotopiemengen und ihre induzierten Abbildungen. I, Math. Z. 69 (1958), 299-344. MR 20 #6698. MR 0100265 (20:6698)
  • [28] M. Rothenberg and J. Sondow, Non-linear smooth representatives of compact lie groups (preprint).
  • [29] H. Sato, Diffeomeophism groups of $ {S^p} \times {S^q}$ and exotic spheres, Quart. J. Math. Oxford Ser. (2) 20 (1969), 255-276. MR 40 #6584. MR 0253369 (40:6584)
  • [30] N. E. Steenrod, The topology of fibre bundles, Princeton Math. Ser., vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. MR 0039258 (12:522b)
  • [31] E. Stone, Ph. D. Thesis, Cornell University, 1972 (to appear).
  • [32] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 15, 890. MR 0061823 (15:890a)
  • [33] C. T. C. Wall, Free piecewise linear involutions on spheres, Bull. Amer. Math. Soc. 74 (1968), 554-558. MR 36 #5955. MR 0222905 (36:5955)
  • [34] -, Surgery on compact manifolds, Academic Press, New York, 1972.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57D65, 57E25

Retrieve articles in all journals with MSC: 57D65, 57E25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0431213-1
Keywords: Free differentiable involutions, Browder-Livesay invariant, equivariant diffeomorphism, concordance group of diffeomorphisms, curious involutions
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society