Questions from students on 2020-02-04

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1. Why do we get a sign change when we do a single transposition? ie
why does ab = -ba \forall a, b \in \calA_1?

Answer: In cohomology, this comes from the "sign" homomorphism from
the symmetric group to \ZZ/2\ZZ.  That is, suppose \Delta is an
n--dimensional simplex.  The symmetric group Sym_{n+1} acts linearly
on \Delta -- the action is induced by permutations of the vertices.
Every time we transpose a pair of vertices we change the "orientation"
of \Delta.  This happens in dimension one as well; you can think of
this as a special case of what happens in general dimension or you can
think of the one-dimensional case as causing the general case.

2. Does homology admit a similar product?  If not, why not?

Answer: The short answer to your first question is "no".  Hatcher
gives one answer to your second question in his introduction to
Chapter 3.  Another answer to your second question might be: "it is
difficult to intersect a pair of homology classes when the ambient
space is singular".  If the ambient space is nice (say a closed finite
dimensional manifold) then it is possible to make a definition;
indeed, this was Poincar\'e's original formulation.  If we get to
this, it will have to be after we prove Poincar\'e duality.