scholarly journals Comparing homotopy categories

2007 ◽  
Vol 2 (1) ◽  
pp. 169-205 ◽  
Author(s):  
David Blanc
Keyword(s):  
Exact Sequence ◽  
Obstruction Theory ◽  
Homotopy Groups ◽  
Model Categories

AbstractGiven a suitable functor T : → between model categories, we define a long exact sequence relating the homotopy groups of any X ε with those of TX, and use this to describe an obstruction theory for lifting an object G ε to . Examples include finding spaces with given homology or homotopy groups.

scholarly journals Obstruction theory in model categories

2004 ◽  
Vol 181 (2) ◽  
pp. 396-416 ◽  
Author(s):  
J.Daniel Christensen ◽  
William G. Dwyer ◽  
Daniel C. Isaksen
Keyword(s):  
Obstruction Theory ◽  
Model Categories

scholarly journals Some results in quasitopological homotopy groups

2020 ◽  
Vol 72 (12) ◽  
pp. 1663-1668
Author(s):  
T. Nasri ◽  
H. Mirebrahimi ◽  
H. Torabi
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Homotopy Group ◽  
Loop Space ◽  
Homotopy Groups

UDC 515.4 We show that the th quasitopological homotopy group of a topological space is isomorphic to th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field.

Brunnian braids over the 2-sphere and Artin combed form

2021 ◽  
pp. 2150012
Author(s):  
Duzhin Fedor ◽  
Loh Sher En Jessica
Keyword(s):  
Exact Sequence ◽  
Word Problem ◽  
Open Problem ◽  
Homotopy Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Homotopy Groups ◽  
Class Groups ◽  
Homotopy Groups Of Spheres

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

scholarly journals Classification of knotted tori

2019 ◽  
Vol 150 (2) ◽  
pp. 549-567
Author(s):  
A. Skopenkov
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Smooth Manifold ◽  
Group Structure ◽  
Extension Problem ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stiefel Manifolds

AbstractFor a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

scholarly journals An Application of the Path-Space Technique to the Theory of Triads

1963 ◽  
Vol 22 ◽  
pp. 169-188 ◽  
Author(s):  
Yasutoshi Nomura
Keyword(s):  
Exact Sequence ◽  
Homotopy Theory ◽  
Path Space ◽  
Homotopy Groups ◽  
Space Technique ◽  
Result Theorem

One of the most powerful tools in homotopy theory is the homotopy groups of a triad introduced by Blakers and Massey in [1]. Our aim here is to develop systematically the formal, elementary aspects of the theory of a generalized triad and the mapping track associated with it. This will be used in §5 to deduce a result (Theorem 5.5) which seems to be closely related to an exact sequence established by Brown [2].

scholarly journals On long exact (π¯,ExtΛ)-sequences in module theory

2004 ◽  
Vol 2004 (26) ◽  
pp. 1347-1361
Author(s):  
C. Joanna Su
Keyword(s):  
Exact Sequence ◽  
Homotopy Group ◽  
Homotopy Theory ◽  
Infinite Sequence ◽  
Homotopy Groups ◽  
Module Theory

In (2003), we proved the injective homotopy exact sequence of modules by a method that does not refer to any elements of the sets in the argument, so that the duality applies automatically in the projective homotopy theory (of modules) without further derivation. We inherit this fashion in this paper during our process of expanding the homotopy exact sequence. We name the resulting doubly infinite sequence the long exact(π¯,ExtΛ)-sequence in the second variable—it links the (injective) homotopy exact sequence with the long exact ExtΛ-sequence in the second variable through a connecting term which has a structure containing traces of both a π¯-homotopy group and an ExtΛ-group. We then demonstrate the nontriviality of the injective/projective relative homotopy groups (of modules) based on the results ofs Su (2001). Finally, by inserting three (π¯,ExtΛ)-sequences into a one-of-a-kind diagram, we establish the long exact (π¯,ExtΛ)-sequence of a triple, which is an extension of the homotopy sequence of a triple in module theory.

Obstructions to Lie–Rinehart Algebra Extensions

2011 ◽  
Vol 18 (01) ◽  
pp. 83-104 ◽  
Author(s):  
J. M. Casas
Keyword(s):  
Exact Sequence ◽  
Obstruction Theory

The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

Closed model categories for the n-type of spaces and simplicial sets

1995 ◽  
Vol 118 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Carmen Elvira-Donazar ◽  
Luis-Javier Hernandez-Paricio
Keyword(s):  
Base Point ◽  
Category Structure ◽  
Model Category ◽  
Homotopy Groups ◽  
Weak Equivalence ◽  
Model Categories ◽  
Closed Model ◽  
Simplicial Sets ◽  
Closed Model Categories ◽  
Closed Model Category

AbstractFor each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

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