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I am struggling to gain a geometric intuition for the concept of homology groups. In particular, my problem is in understanding the algebraic terms "kernel" and "quotient" and connect them with boundaries of simplices, holes etc.

Definitions:

For a simplicial complex $X$, the $n^{th}$ homology $H_n(X)$ is $Z_n/B_n$, where $Z_n = \ker(d_n : C_n \to C_{n-1})$ is the group of cycles and $B_n = \text{im}(d_{n+1} : C_{n+1} \to C_n)$ is the group of boundaries.

Question 1: If kernel is defined as elements that are mapped to the group identity, what exactly is the identity in $C_n$? Is that some "identity chain" in $C_n$, that is, something like a trivial formal sum?

Question 2: How to see the "cycles modulo boundaries" in the "$Z_n/B_n$"? I visualize homology as describing holes = objects without a boundary that are not a boundary of something else. But how does the quotient correspond to this? Maybe I am struggling with the concept of quotient, but I understand it such as "$A/B$" means that elements in $A$ are glued together to elements in $B$. I just cannot connect this to the geometric intuition about homology.

Thank you for your help.

EDIT: I am studying mostly Hatcher´s Algebraic Topology (my university course also follows this book) and additionaly I use Wikipedia. Also, I have already read many posts on "(co)homology intuition" here, I am really only asking these particular questions. Thank you.

Tereza Tizkova
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  • I never understood any of the geometric intuition behind this stuff until I read Henle's old book on combinatorial topology. – Randall Aug 22 '22 at 13:44
  • @Randall I understand either the geometry, or the algebra (separately), but fail to understand them together haha. And I read Hatcher´s Algebraic Topology. – Tereza Tizkova Aug 22 '22 at 14:39
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    You might check out https://math.stackexchange.com/questions/4444397/homology-and-homotopy-groups-and-holes/4446074#4446074 – Cheerful Parsnip Aug 22 '22 at 14:48
  • The (additive) identity is the zero chain. Thus, the subgroup of $n$-cycles consists of $n$-chains with zero boundary. As for the geometric intuition: This was discussed at MSE numerous times. My suggestion is to work out simple examples of 1-dimensional simplicial complexes to see the geometric meaning of the definition. – Moishe Kohan Aug 22 '22 at 15:00
  • Yes, a trivial formal sum. 2. Yes, you struggle with quotient. $A/B$ is precisely elements of $A$ modulo $B$.
    • – Арсений Кряжев Aug 22 '22 at 15:00
  • @CheerfulParsnip Thank you, however I have seen that and many other posts here. My problem is mainly the two questions. The problem is the quotient and the kernel, not overall intuition. – Tereza Tizkova Aug 22 '22 at 15:02
  • @АрсенийКряжевiswithUkraine Thanks! However, I am struggling with how the trivial formal sums modulo image of $d_{n+1}$ correspond geometrically to "objects without a boundary that are not boundary themselves" ... Like HOW exactly does the $Z_n/B_n$ look, how is it "glued together", how does the quotient there work geometrically? – Tereza Tizkova Aug 22 '22 at 15:04
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    "objects without a boundary" is $Z_n$ and "that are not boundary themselves" is $/B_n$. A quotient group $G/H$ is the result of taking $G$ and proclaiming that everything in $H$ is zero. This will force some relations, e.g. $g + h = g,$ and the result will be what strictly is defined as the group of cosets. Thus, $Z_n/B_n$ is "let's look at cycles and proclaim that boundaries don't count." – Арсений Кряжев Aug 22 '22 at 15:11
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    @АрсенийКряжевiswithUkraine Ah, I think I finally see it! Thank you a lot! So $Z_n/B_n$ is something like "omitting the n-dimensional objects that are a boundary of something (n+1)-dimensional and considering only the other n-dimensional objects, but such that they are trivial formal sums at the same time." This starts to give the intuition on why these objects are the "empty areas" or "holes" in the space. – Tereza Tizkova Aug 22 '22 at 15:22