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Real Computable Manifolds and Homotopy Groups
[chapter]

2009
*
Lecture Notes in Computer Science
*

We state definitions of

doi:10.1007/978-3-642-03745-0_16
fatcat:ktkwh4gkxze6lcke6dolxv5jte
*real*-*computable**manifold**and*of*real*-*computable*paths in such*manifolds*,*and*show that, while BSS machines cannot in general decide such questions as nullhomotopy*and*simple connectedness ... for such structures, there are nevertheless*real*-*computable*presentations of paths*and**homotopy*equivalence classes under which such*computations*are possible. ... In [3] , Brown proved that it was possible, from a presentation of a*manifold*as a finite simplicial complex, to*compute*a presentation for each of the*homotopy**groups*of the*manifold*, including π 1 . ...##
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Page 1346 of Mathematical Reviews Vol. 47, Issue 5
[page]

1974
*
Mathematical Reviews
*

Here he

*computes*the stable*homotopy**groups*of the stunted*real*projective space 7,,,(P**+%/p*-+) (k2>p+2) for p<22. ... Then the well-known isomorphism T+ p(\Viewa.4) = 74 p(P*t3/P*-1) for k2p+2 gives the*homotopy**groups*7; 4 5(V;.44,4) of the*real*Stiefel*manifold*Vie+a,4 for p<22. ...##
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Page 1170 of Mathematical Reviews Vol. , Issue 86c
[page]

1986
*
Mathematical Reviews
*

A (6) 3 (1984), no. 3, 401-410.
86c:57035

*MANIFOLDS**AND*CELL COMPLEXES 1170 In this note, the author proves the following result: Let G be a noncompact*real*algebraic*group*,*and*let c € H,(G, Z2); then ... The author*computes*the tangent bundles of smooth*homotopy*lens spaces which arise as quotients of Brieskorn spheres by free actions of finite cyclic*groups*. ...##
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Page 4652 of Mathematical Reviews Vol. , Issue 90H
[page]

1990
*
Mathematical Reviews
*

He also determines the I’-

*group*I,,,3(X)*and*obtains a complete algebraic*homotopy*invariant for A‘-polyhedra with torsion-free integral homology (n > 3),*and*as its application he*computes*the*group*of ... In §3 we form the bordism*groups*of R-*manifolds**and*in §4 we show that there is a duality of Poincaré type between the bordism*groups**and*the cobordism*groups*with reality.” 90h:55009 55N22 57R77 Ossa, ...##
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A Topological Obstruction to the Removal of a Degenerate Complex Tangent and Some Related Homotopy and Homology Groups
[article]

2015
*
arXiv
*
pre-print

We further

arXiv:1506.07985v1
fatcat:772ua3yp2jgipc2n5k2jasuvva
*compute*additional*homotopy**and*homology*groups*for the space Y*and*of its complement W consisting of "partially complex" 3-planes in C^3. ... The obstruction is a certain*homotopy*class of the space Y consisting of totally*real*3-planes in the Grassmanian of*real*3-planes in C^3 (=R^6). ... The*Homotopy**Groups*of Y In this section, we*compute*the first two*homotopy**groups*of the space Y consisting of totally*real*3-planes in G 6,3 . ...##
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A separable manifold failing to have the homotopy type of a CW-complex
[article]

2006
*
arXiv
*
pre-print

We show that the Pr\"ufer surface, which is a separable non-metrizable 2-

arXiv:math/0609665v1
fatcat:z2tdzw6ofve7pfgcf6vijrgsza
*manifold*, has not the*homotopy*type of a CW-complex. This will follow easily from J. H. C. ... Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a*homotopy*equivalence, then the given space is not*homotopy*equivalent to a CW-complex. ... I am very much obliged to Matthew Baillif for precious guidance through some theory of non-metrizable*manifolds*,*and*to André Haefliger for suggesting the method to*compute*π i (R). ...##
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A family of cohomological complex projective spaces
[article]

2018
*
arXiv
*
pre-print

We exhibit a family of complex

arXiv:1810.09029v1
fatcat:4qkabruagbfjvmbbklpbgzeqsq
*manifolds*, which has a member at each odd complex dimension*and*which has the same cohomology*groups*as the complex projective space at that dimension, but not*homotopy*equivalent ... The first author would like to thank to his father*and*family for their support during the preparation of this paper. This work is partially supported by the grant ♯114F320 of Tübitak 1 . ... Considering the*homotopy**groups*, the Stiefel*manifold*V 2 R 2k is 2k-3 connected*and*the 2k-2-nd*homotopy**group*of it is Z. ...##
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Page 9119 of Mathematical Reviews Vol. , Issue 2001M
[page]

2001
*
Mathematical Reviews
*

In addition, the fundamental

*group*of each*manifold*of this family is*computed*in this section. Finally, an ar- gument from [B. Ozbagci*and*A. I. ...*group*. ...##
###
Page 7992 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

The subset of

*real*- ized obstructions is denoted BL( ¥, X)*and*those realized by simple*homotopy*equivalences with trivial normal invariant by B( ¥, X )— they can be described via the Ranicki assembly ... /a, a is equivalent to a based free action f of G on Q if*and*only if Qo/f admits the structure of a Q-*manifold**and*Qo/P has the proper*homotopy*type of Qo/a. ...##
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Page 3255 of Mathematical Reviews Vol. , Issue 92f
[page]

1992
*
Mathematical Reviews
*

Summary: “We

*compute*the stable*homotopy**groups**and*a few of the unstable*homotopy**groups*of the homogeneous spaces SO(2n)/U(s)*and*U(2n)/Sp(s) for s < n,*and*those of SO(4n)/Sp(s) for s <n. ... Suppose that a locally compact*group*G acts via principal bundle automorphisms of a continuous principal H-bundle P — M, where H is a*real*reductive*group**and*M is separable*and*metrizable. ...##
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Topology in the 20th century: a view from the inside

2004
*
Russian Mathematical Surveys
*

The classification of

doi:10.1070/rm2004v059n05abeh000770
fatcat:3htmvalrnbhvfhxgqg2dff2pgm
*manifolds*reduces to the*computation*of a single*homotopy**group*of the "Thom space of the normal bundle". ... As we have said, homology*and**homotopy**groups*are*homotopy*invariant. ...##
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Page 3708 of Mathematical Reviews Vol. , Issue 84i
[page]

1984
*
Mathematical Reviews
*

For 2,(X)=Zy' (=Z,XZ,X --- XZ, with m factors)

*and*the various w( X), the surgery obstruction*groups*L,(7,(X),w(X)) have been*computed*by Wall, Bass*and*the author. ... Tognoli (Pisa) 57S _ Topological transformation*groups*See also 14031, 53037. Balcerak, W.; Hajduk, B.*Homotopy*type of automorphism*groups*of*manifolds*. Collog. Math. 45 (1981), no. 1, 1-33 (1982). ...##
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Book Review: Rings, modules, and algebras in stable homotopy theory

2000
*
Bulletin of the American Mathematical Society
*

By the mid 1960's, Adams had pursued spectral sequence

doi:10.1090/s0273-0979-00-00899-5
fatcat:tp7h6o4by5bb7on22msulbnf3m
*computations*of the stable*homotopy**groups*of spheres using K-theory,*and*Novikov [21] developed the analogous*computations*using complex cobordism ... Each choice gives rise to a spectrum MG, whose*homotopy**groups*are isomorphic to the graded ring of cobordism classes of*manifolds*with structure*group*in the chosen family. ...##
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Page 4381 of Mathematical Reviews Vol. , Issue 82j
[page]

1982
*
Mathematical Reviews
*

The author considers control systems of the form (+) x=f(x)+ug(x), where f, g are

*real*-analytic vector fields on a connected,*real*-analytic two-dimensional*manifold*M*and*u is a*real*-valued control. ... Rigdon, Robert; Williams, Bruce 82j:57031 Embeddings*and*immersions of*manifolds*. Geometric applications of*homotopy*theory ( Proc. ...##
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On rational homotopy of four-manifolds
[article]

2003
*
arXiv
*
pre-print

We give explicit formulas for the ranks of the third

arXiv:math/0309076v1
fatcat:ecvyt3whujcmrg4ownrkwsal54
*and*fourth*homotopy**groups*of all oriented closed simply-connected four*manifolds*in terms of their second Betti numbers. ... We also show that the rational*homotopy*type of these*manifolds*is classified by their rank*and*signature. ...*Computation*of the ranks of the*homotopy**groups*. Before we proceed to the*computation*of the ranks of the low degree*homotopy**groups*of four-*manifolds*, let us note the following important facts. ...
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