Intereting Posts

gradient descent optimal step size
How to prove the quotient rule?
Can the factorial function be written as a sum?
Method of characteristics for the PDE $xu_x +y u_y = 0$ with an initial condition on a circle
Manifold notes in more informal way
Proving a function is onto and one to one
Is R with $j_d$ topology totally disconnected?
How does one prove that the Klein bottle cannot be embedded in $R^3$?
convergence of entropy and sigma-fields
Definition of uniform structure
an exercise problem in the book “linear algebra” by Kenneth Hoffman and Ray Kunze, pg #5, Q.2
When does the integral preserve strict inequalities?
construction of a linear functional in $\mathcal{C}()$
Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$
Sum of positive definite matrices still positive definite?

I’ve seen this but didn’t really understand the answer. So here is what I tried:

- A short exact sequence that cannot be made into an exact triangle. (Weibel 10.1.2)
- Finite injective dimension
- Homology of a simple chain complex
- Description of $\mathrm{Ext}^1(R/I,R/J)$
- Positivity of the alternating sum associated to at most five subspaces
- Cohomological dimension of direct product

According to this picture we have one 0-simplex – $[v]$, two 1-simplices – $[v,v]_a,[v,v]_b$ and two 2-simplices – $[v,v,v]_U,[v,v,v]_L$.

**The chain complexes**

$C_0=\{nv:n\in\Bbb{Z}\}, C_1=\{n[v,v]_a+m[v,v]_b+k[v,v]_c:n,m,k\in\Bbb{Z}\}, C_2=\{n[v,v,v]_U+m[v,v,v]_L:n,m\in\Bbb{Z}\}$.

**Boundary maps**

$∂_0=0, ∂_1([v,v]_a)=[v]-[v]=0$,

$∂_2([v,v,v]_U)=[v,v]_a+[v,v]_b-[v,v]_c=[v,v]_a$ and $∂_2([v,v,v]_L)=[v,v]_a-[v,v]_b+[v,v]_c=[v,v]_a$.

I think $\operatorname{Im}∂_2$ is the set of multiples of $[v,v]$ which is isomorphic to $\Bbb{Z}$ but I’m not sure about this. Once I know $\operatorname{Im}∂_2$ it’s easy to calculate $\ker{∂_1}/\operatorname{Im}∂_2=H_1(K)$ as $\ker{∂_1}=\Bbb{Z}^3$.

Could you explain what is $\operatorname{Im}∂_2$?

- Kähler differential over a field
- On the bounded derived category of a finite dimensional algebra with finite global dimension
- Can it happen that the image of a functor is not a category?
- Can we think of a chain homotopy as a homotopy?
- Given relations on matrices $H,V,$ and some vectors, can we deduce that $x = 0$?
- Signs in the tensor product and internal hom of chain complexes
- Bockstein homomorphism for $\mathbb{Z}_n$ and “Steenrod” $n$th power
- What are exact sequences, metaphysically speaking?

You obviously have a typo, having forgotten to include the $1$-simplex $c$. Your “simplifications” of $\partial_2$ are incorrect. To save typing, I’ll write $\partial_2 U = a+b-c$ and $\partial_2 L= a-b+c$. If you make the change of basis (over $\mathbb Z$) $a’=a+b-c$, $b’=b-c$, $c’=c$, you can easily check that $a’, a’-2b’$ give a basis for $\text{im}\,\partial_2$. So $H_1 \cong \mathbb Z\oplus \mathbb Z/2$.

It seems it is a long way from being accepted that to write the boundary of the diagram

we need write only the **nonabelian** formula (assuming base point the top left corner)

$$\delta \sigma= b +a -b +a $$

saving all that decomposition into simplices. For more information see my presentation at Chicago, 2012. It is also true that it does require new ideas to make a complete story, like crossed modules, free crossed modules ($\sigma$ is a generator of a free crossed module, and $a,b$ generate a free group), and the complete story is in the book “Nonabelian algebraic topology”, 2011, EMS Tract 15. Other presentations (2014, 2015) are on my preprint page (Galway, Aveiro, Liverpool,…). All this stems from and develops the work of J.H.C. Whitehead in his paper “Combinatorial Homotopy II”.

- Intuition – Fundamental Homomorphism Theorem – Fraleigh p. 139, 136
- Conditional Expectation of Functions of Random Variables satisfying certain Properties
- Permutations on word $MISSISSIPPI$.
- $x^3+48=y^4$ does not have integer (?) solutions
- Is finding the length of the shortest addition chain for a number $n$ really $NP$-hard?
- Commutativity of “extension” and “taking the radical” of ideals
- How many points must the arc intersect?
- To show that the set point distant by 1 of a compact set has Lebesgue measure $0$
- How can i find closed-form expression of generating function of this series?
- Is this proof of the fundamental theorem of calculus correct?
- Example of a normal extension.
- problem on uniform convergence of a riemann integrable function
- Fibonacci numbers and proof by induction
- Division of $q^n-1$ by $q^m-1$, in Wedderburn's theorem
- Evaluate $\cos 18^\circ$ without using the calculator