3

This answer might be overkill considering that the body of OP's question only asks for answers to 4 subproblems, but for completeness it might be worth it to spell out the solution to the whole question in the subject. (I recommend you first try to work out for yourself the hints in Thomas Rot's comment and then put the results together, because this answer ...


2

The map is surjective since all generators are in the image. Since each grading is a finite set this implies that it is bijective, and so it is an automorphism.


1

Of course your MV answer would work eventually, but you may use cellular homology as well. Pick a cellular structure on $X$, say $X^{0}=\mbox{point}$, $X^{1}=X^{0}$, $X^{2}=S^{2}\vee S^{2}$, $X^{3}=X^{2}\cup D^{3}$, $X^{4}=X$. Then $X^{4}/X^{3}=S^{4}$, $X^{3}/X^{2}=S^{3}$, $X^{2}/X^{1}=S^{2}\vee S^{2}$, $X^{1}/X^{0}=\mbox{point}$, and the cellular homology ...


1

In general, suppose that you have a piece of an exact sequence like this: $$A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{j} E $$ and suppose also that you know $A$, $B$, $D$ and $E$, and you know the maps $f$ and $j$. Then you can immediately deduce that $C$ fits into a "short" exact sequence like this: $$0 \mapsto \text{coker}(f) \...


1

Hint : Use that $H_n(S^n\vee S^n) \cong H_n(S^n)\oplus H_n(S^n)$, the isomorphism being induced by the projections $S^n\vee S^n\to S^n$ : then what does $S^n\to S^n\vee S^n\to S^n$ look like ?


1

I like to think of the formula for $h\sigma$ as a description of a subdivision of the simplicial cylinder $\sigma \times [0,1]$ as a simplicial complex. The equation $h \circ \partial + \partial \circ h = \text{Id} - s$, when rewritten as $$\partial \circ h = \text{Id} - s - h \circ \partial $$ can be understood geometrically like this: The term $\text{...


1

An orientation of an $n$-simplex $\sigma \in \cal N$ is a choice of one of two equivalences classes of orderings of its vertex set, where two orderings are equivalent if the re-ordering map is an even permutation. For example, letting $\Delta^n$ denote the "standard" rectilinear $n$-simplex in $\mathbb R^n$, with vertex set $0,p_1,...,p_n$ where $p_i$ is ...


1

The spectral sequence comment outlined in the comments is sufficient. When $G$ is connected, use the Hopf theorem that $H^*(G;\Bbb Q) \cong \Lambda(x_{k_i})$, the exterior algebra on some number of generators $x_{k_i}$ with odd degree. Then the cohomological Serre spectral sequence applied to $BG \to EG \to G$ forces $BG$ to be a polynomial algebra on ...


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