Queer questions about topology

[navigator]

Many abstract mathematical concepts have their intuitive correspondences or geometrical meanings. such as differentiable is corresponding to "smooth", determinant is corresponding to "volumn",homolgy group is corresponding to "hole".

1.The question is whether "exact" and "exact sequence" have their intuitive correspondence? Do they mean "no any hole"(acyclic) or "no gap" or "no split"(split exact sequence) or something else?what is the geometrical meaning about "exact differential"?

2.what is the intuitive meaning of the word "flat" in "flat module"? Mathematician use "exact functor" to define "flat module",and whether this imply that "exact" relate with the intuitive concept "flat"?

3."Cutting" and "gluing" are the basic operations of topology,and "gluing" is corresponding to quotient topology,then what is "cutting" corresponding to? product topology? "cutting along B" is corresponding to $$B \times \partial I$$ namely "Bx{0},Bx{1}"?

4.Why the mathematician call identification topology "quotient topology"?For instance,one could get cylinder from rectangle via identification IxI/~(where (0,y)~(1,y)),and we may see that the operation is like to subtract the edge ((0,y) or (1,y)) from the rectangle,so identification should bear an analogy with subtraction，rather than division.Further more,subspace topology is dual to quotient topology,direct sum topology is dual to product topology,then what is the relation between quotient topology and product topology? inversion or other?

Queer question,huh~~~

 

[matt grime]

1. If you look at homology, etc, then exact means 'no homology' which is 'no holes'.

2. Flat means X\otimes - is exact.

3. I don't buy that 'gluing' is the quotient topology, and cutting is not the product topology. I think your analogies are bad here.

4. Seems like more semantics as in 3.

 

[mathwonk]

geometrically, a map of regular rings is flat if the dimensions of all the fibers of the map are the same.

i.e. a flat map of smooth varieties is one whose fiber dimensions are constant

this is an attempt to relate "flat" somehow to geometry and constancy, but i recall my algebra teacher (matsumura) laughed when trying to justify the term flat geometrically this way.

 

[navigator]

Thanks for reply. My questions may be naive,for my major is not mathematics,but when i read the articles that i want to learn,there appear many topology term,and the modern mathematics is so abstract to me,so i has to use some falsework to help me to reach the mathematics building in my mind. Question still here is how could we get the formal mathematical definitions of the intuitive operation "cutting" and "gluing"? relate to the surgery theory?

 

[matt grime]

I think you might want to know about excision, and surgery. Gluing is a pushout diagram.

If I have inclusions X to Y and X to Z I can form the space Y u_X Z of the pushout diagram.

 

[navigator]

Then could we say cutting is a pullback diagram? Pullback is dual to pushout,and what is the inversion of pushout? what is difference between dual and inversion.I once had a naive idea: glue=sum,paste=quotient(identication),cut=product,puncture(delete)=difference(excision),and surgery theory combine them all,but it seem to be improper now~~~