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Real Computable Manifolds and Homotopy Groups [chapter]

Wesley Calvert, Russell Miller
2009 Lecture Notes in Computer Science  
We state definitions of 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 .  ... 
doi:10.1007/978-3-642-03745-0_16 fatcat:ktkwh4gkxze6lcke6dolxv5jte

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.  ... 

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.  ... 

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,  ... 

A Topological Obstruction to the Removal of a Degenerate Complex Tangent and Some Related Homotopy and Homology Groups [article]

Ali M. Elgindi
2015 arXiv   pre-print
We further 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 .  ... 
arXiv:1506.07985v1 fatcat:772ua3yp2jgipc2n5k2jasuvva

A separable manifold failing to have the homotopy type of a CW-complex [article]

Alexandre Gabard
2006 arXiv   pre-print
We show that the Pr\"ufer surface, which is a separable non-metrizable 2-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).  ... 
arXiv:math/0609665v1 fatcat:z2tdzw6ofve7pfgcf6vijrgsza

A family of cohomological complex projective spaces [article]

Mustafa Kalafat, Dersim Kaya
2018 arXiv   pre-print
We exhibit a family of complex 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.  ... 
arXiv:1810.09029v1 fatcat:4qkabruagbfjvmbbklpbgzeqsq

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.  ... 

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.  ... 

Topology in the 20th century: a view from the inside

S P Novikov
2004 Russian Mathematical Surveys  
The classification of 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.  ... 
doi:10.1070/rm2004v059n05abeh000770 fatcat:3htmvalrnbhvfhxgqg2dff2pgm

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).  ... 

Book Review: Rings, modules, and algebras in stable homotopy theory

John McCleary
2000 Bulletin of the American Mathematical Society  
By the mid 1960's, Adams had pursued spectral sequence 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.  ... 
doi:10.1090/s0273-0979-00-00899-5 fatcat:tp7h6o4by5bb7on22msulbnf3m

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.  ... 

On rational homotopy of four-manifolds [article]

S. Terzic
2003 arXiv   pre-print
We give explicit formulas for the ranks of the third 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.  ... 
arXiv:math/0309076v1 fatcat:ecvyt3whujcmrg4ownrkwsal54
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