Notes from Craigfest - University of Melbourne
... 3. If {Xi }i∈I is a set of based spaces, then the inclusions Xi −→ ∨l∈I Xl induces an isomorphism π(ji )∗ : πi∈I Ẽ q (Xi ) ∼ = Ẽ q (∨l∈I Xl ). 4. If f : X → Y is a weak equivalence (ie. an isomorphism in πq for all q) then f ∗ : Ẽ ∗ X → Ẽ ∗ Y is an isomorphism. Example: H ∗ (−; A), singular coho ...
... 3. If {Xi }i∈I is a set of based spaces, then the inclusions Xi −→ ∨l∈I Xl induces an isomorphism π(ji )∗ : πi∈I Ẽ q (Xi ) ∼ = Ẽ q (∨l∈I Xl ). 4. If f : X → Y is a weak equivalence (ie. an isomorphism in πq for all q) then f ∗ : Ẽ ∗ X → Ẽ ∗ Y is an isomorphism. Example: H ∗ (−; A), singular coho ...
Math 261y: von Neumann Algebras (Lecture 1)
... Passing from the group G to the von Neumann algebra of a representation α : G → B(V ) generally loses a great deal of information. However, it retains the information we are interested in: namely, the structure of all G-equivariant direct sum decompositions of G. Moreover, the von Neumann algebra o ...
... Passing from the group G to the von Neumann algebra of a representation α : G → B(V ) generally loses a great deal of information. However, it retains the information we are interested in: namely, the structure of all G-equivariant direct sum decompositions of G. Moreover, the von Neumann algebra o ...
Algebraic closure
... PROOF. The idea of the proof is simple: consider all fields (E, +, · ) which are algebraic extensions of F , find a maximal one among them by Zorn’s Lemma, and show that it is algebraically closed by virtue of having no further algebraic extensions. There are two technical complications. First of al ...
... PROOF. The idea of the proof is simple: consider all fields (E, +, · ) which are algebraic extensions of F , find a maximal one among them by Zorn’s Lemma, and show that it is algebraically closed by virtue of having no further algebraic extensions. There are two technical complications. First of al ...
An algebraic topological proof of the fundamental theorem of al
... Example 1.1:(linear homotopy) Any two paths in Rn having same initial and final points are homotopic via the homotopy ft (s) = (1 − t)f0 (s) + tf1 (s). We can see from the above definition that the homotopy of paths is an equivalence relation in the set of paths from I → X. The equivalence class of ...
... Example 1.1:(linear homotopy) Any two paths in Rn having same initial and final points are homotopic via the homotopy ft (s) = (1 − t)f0 (s) + tf1 (s). We can see from the above definition that the homotopy of paths is an equivalence relation in the set of paths from I → X. The equivalence class of ...
On Colimits in Various Categories of Manifolds
... So far we’ve shown that Mn f d is poorly behaved with respect to even very nice pushouts and coequalizers. A simple example to show it’s poorly behaved with respect to direct limits is the sequence R → R2 → R3 → . . . , where the colimit R∞ is not a manifold. The only colimits of interest left are t ...
... So far we’ve shown that Mn f d is poorly behaved with respect to even very nice pushouts and coequalizers. A simple example to show it’s poorly behaved with respect to direct limits is the sequence R → R2 → R3 → . . . , where the colimit R∞ is not a manifold. The only colimits of interest left are t ...
On the Choquet-Dolecki Theorem
... then x ∈ Φ(t0 ) Theorem 1.2 has subsequently been refined in terms of the following definitions. A space X is angelic [8] if (i) each relatively countably compact subset (i.e. every sequence of distinct elements of the set has a cluster point) of X is compact; (ii) each point in the closure of a rela ...
... then x ∈ Φ(t0 ) Theorem 1.2 has subsequently been refined in terms of the following definitions. A space X is angelic [8] if (i) each relatively countably compact subset (i.e. every sequence of distinct elements of the set has a cluster point) of X is compact; (ii) each point in the closure of a rela ...
![arXiv:1705.08225v1 [math.NT] 23 May 2017](http://s1.studyres.com/store/data/017200628_1-6512c4c34735fe39f7f1d2a26e5283a8-300x300.png)
arXiv:1705.08225v1 [math.NT] 23 May 2017
... T (g) = α∗′ ○ α∗ ∶ JK → JK ′ . Note that the homomorphism T (g) is a priori defined over the base field of MK∩gK ′g−1 , but its differential at the origin maps the tangent space Ω1 (M K ) into Ω1 (M K ′ ), hence it is defined over Q. We therefore get a map ρJ ∶ T̃K,K ′ → Hom(JK , JK ′ ). Since Hom(J ...
... T (g) = α∗′ ○ α∗ ∶ JK → JK ′ . Note that the homomorphism T (g) is a priori defined over the base field of MK∩gK ′g−1 , but its differential at the origin maps the tangent space Ω1 (M K ) into Ω1 (M K ′ ), hence it is defined over Q. We therefore get a map ρJ ∶ T̃K,K ′ → Hom(JK , JK ′ ). Since Hom(J ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.