If c top and w is the homotopy equivalences, then cw 1 is isomorphic to the category with. Let be the inclusion map and be the constant map we need to find such that and define exactly the same formula works for example 2. Reading allen hatchers book available online via this link on algebraic topology, it states on page 3 that homotopy type defines an equivalence relation. Msuch that fg and gf are homotopic to the identity morphisms of n and m respectively.
In particular youre confused about which direction is easy. In particular, it is proved that differentiably homotopic maps induce. Introduction to higher homotopy groups and obstruction theory. The homotopy extension property this note augments material in hatcher, chapter 0.
In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. First, there is the notion of chain homotopy equivalence, which means having an inverse up to chain 1424 noticesofthe american mathematical society volume 66,number 9. This is the terminology common for instance in the standard proof of the poincare lemma. We shall use the symbol xny to mean that x and y are of the same whomotopy type i. A chain homotopy is a homotopy in a category of chain complexes with respect to the standard interval object in chain complexes. Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence.
In the last lecture we introduced relative singular homology of a pair of spaces. One might expect that if a is contractible, the quotient map q. Notes on tor and ext 5 which is an isomorphism if l and l. In fact, its homotopy category is equivalent to the derived category dr, formed by taking chr modulo the equivalence relation given by. Homotopy algebra of openclosed strings 231 other descriptions by multivariable operations and coderivation differentials. We assume throughout that a is a closed subspace of x. Being homotopic is an equivalence relation, so we have equivalence classes. Xf and x y will mean that x and y are of the same homotopy type. Let r be a ring and chr the category of chain complexes of rmodules.
For each point x2x, let e xdenote the constant map i. Math 9052b4152b algebraic topology winter 2015 homology. Is it true for chain complexes of free abelian groups. First h f g k h 1y k h k 1z similarly we have the other direction.
Mar 11, 2015 this feature is not available right now. Equivalence classes are referred to as homotopy classes of maps. Let c be the cantor set with the discrete topology. Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. X the xcontrolled categories of ranicki and weiss 7. If fis homotopic to gvia hand gis homotopic to hvia k, then fis homotopic to. Equivariant stable homotopy theory 5 isotropy groups and universal spaces. If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent. Homotopy theory in a model category let x and y be.
This is the quotient of xby an equivalence relation, where xyif there exists a continuous path p. Sometimes a chain homotopy is called a homotopy operator. The subdivision chain equivalence on simplicial chains induced. We define a refinement of the homotopy relation which is generated by composing maps with homotopy trivial self equivalences of the objects. There are ways to deal with this, generally along the following lines. In a more physically oriented paper 38, we gave an alternative interpretation in the language of. Its morphisms are maps of complexes modulo homotopy. This is a fancy name for the equivalence class of homotopic functions that has 0 as its canonical representative. The homotopy extension property not all inclusions a. K k such th at gf 1 k and fg 1 k where 1 k and 1 k are the identity 4,2 chain maps. The symmetry and reflexiveness are immediately seen, but the transitivity requires a little work.
An isomorphism of this category is called a chain homotopy equivalence. Homotopy and the fundamental group city university of. But, as before, chain complexes of projective modules are also quite general. We have aan abelian category, and so we know that cha is an abelian. To any topological space xequipped with a base point x. Cinduces isomorphisms on all homotopy groups, but it is not a homotopy equivalence, so the cw hypothesis is required. His main theorem, then asserts that the homotopy category is equivalent to the full subcategory of. In good cases, every functor category cd is also a model category. Therefore chain homotopic is an equivalence relation. Homeomorphisms, homotopy equivalences and chain complexes. For instance, this is true for chain complexes over a field which are all homotopy equivalent to their homology.
Chaim maps can be clumped together into equivalence classes, based on homotopy. A homotopy equivalence is similarly defined in terms of ordinary homotopy. Homotopy is an equivalence relation on maps from x to y. Pl homeomorphism or chain isomorphism a an additive category a. Chain homotopies tensor product of chain complexes let r be a commutative ring with unity, and let c and d be chain complexes of rmodules e. Doing homotopy theory with 2category theory 3 one problem with this is that the homsets of cw 1 may no longer be small even if those of c are. Math 6510 homework 7 tarun chitra april 30, 2011 2. In 1932 baer studied h2g,a as a group of equivalence classes of extensions. In general, this new equivalence relation on maps does. Thus, to get an equivalence relation on topological spaces, we need to take a symmetric transitive closure. Chain homotopy is an equivalence relation and compositions of chain homotopic maps are chain homotopic so the chain complexes and chain homotopy classes form a category. K k such th at gf 1 k and fg 1 k where 1 k and 1 k are the identity 4,2chain maps. Let a be the class of all connected spaces, each of which is domi.
C 0 be chain homotopy inverses of and 0respectively. When is a quasiisomorphism necessarily a homotopy equivalence. Any nonnegatively graded chain complex has a projective resolution, namely, a quasiisomorphic chain. A chain homotopy equivalence is not chain map with an inverse. Another fundamental reason why we want to treat homotopic maps as being equal is given by the following.
We say that spaces are weak homotopy equivalent topological spaces if they are in the same equivalence class under the equivalence relation thus obtained. It was in 1945 that eilenberg and maclane introduced an algebraic approach which included these groups as special cases. D the equivalences classes of are called chain homotopy classes. Homotopy equivalences and free modules 95 a degree argument now shows that any homotopy equivalence f inducing the identity on rl must be homotopic to the identity. The homotopy category of ncomplexes is a homotopy category. The homotopy category of chain complexes ka is then defined as follows. The idea is you have a special class of maps wcalled the weak equivalences, and these generalize the homotopy equivalences above. Chain contractable a chain map is chain contractable if it is homotopic to the zero chain map. Left homotopy of maps x y is an equivalence relation if x is cofibrant. First as a warmup let us show that the map induces an isomorphism on homology.
Each chain map f is homotopic to itself with chain homotopy h 0. Maurercartan moduli and theorems of riemannhilbert type. Let x,y be two topological spaces, and a a subspace of x. The chain homotopy relation defines an equivalence relation on the set of.
Press question mark to learn the rest of the keyboard shortcuts. So we shall restrict our attention primarily to c diff and c pl. Y is a weak equivalence iff it is a homotopy equivalence. Being chain homotopic is an equivalence relation on chain maps. We say that f and g are chain homotopic or just homotopic if there is such a chain homotopy. T0pm is a homotopy equivalence 3, 18 essentially because topnpln toppl for n 5. Since 2s3r 0, we can assume that f is the identity on the 2skeleton. In mathematical logic and computer science, homotopy type theory hott h. Simplicial homotopy theory, link homology and khovanov homology.
By construction, it follows that we have the following relations for the faces of sn. It follows from these definitions that a space x is contractible if and only if the identity map from x to itselfwhich is always a homotopy equivalenceis nullhomotopic. Cellular approximation of topological spaces 93 11. I to y is called homotopy relative to a if for each a in a the map fa,t is constant independent of t. For example, if x and y are homotopy equivalent spaces, then. Spectrum objects in bounded chain complexes are equivalent to unbounded chain complexes ch. Homeomorphisms,homotopy equivalences and chain complexes. The doldkan correspondence generates an equivalence of homotopy categories between spt sab and chr. X is a homeomorphism, and thus a homotopy equivalence. Homotopy equivalence is an equivalence relation on topological spaces. A simplifying point in producing a homotopy simplicial object in relation to a.
The relation of chain homotopy is an equivalence relation on the hom sets of this category. We show for a pyramid y of length r that the associated chain complex of ty is homotopy equivalent to the r1fold. Under what circumstances is a quasiisomorphism between two complexes necessarily a homotopy equivalence. Homotopy equivalences and free modules sciencedirect.
And homotopy groups have important applications, for example to obstruction theory as we will see below. We need to verify that is re exive, symmetric, and transitive. Ferry geometric chain modules are a key ingredient in the following controlled version of browders. X hctslare path homotopic write a p b if a, b have same endpoints p, q. Furthermore, the second author hom14 showed that, modulo an appropriate equivalence relation, the set of knot floer complexes forms a group, and that there is a homomorphism from the knot concordance group to this group. This notion is formalized in the axiomatic definition of a model category a model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. Weak homotopy equivalence of topological spaces topospaces.
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