We prove several theorems about the pseudofree, locally linear and homologically trivial action of finite groups $G$ on closed, connected, oriented 4-manifolds $M$ with non-zero Euler characteristic $(\chi(M) \neq 0)$. In this setting, the $\operatorname{rank}_p(G) \leq 1$, for $p \geq 5$ prime and $\operatorname{rank}(G) \leq 2$, for $p=2,3$.

1. If a non-trivial finite group $G$ acts on $M$ in the above way, then $b_1(M)=0$, and if $b_2(M) \geq 3$, then $G$ must be cyclic and acts semi-freely.
2. If $b_2(M)=2$ and $G=\mathbb{Z}_2 \times \mathbb{Z}_2$, then $M$ must have the same 2-local homology and intersection form as $S^2 \times S^2$. If $G=\mathbb{Z}_q \rtimes \mathbb{Z}_2, q$ odd, is non-abelian, then $M$ must have the same $q$-local homology as $S^2 \times S^2$.
3. If $b_2(M)=1$ and $G=\mathbb{Z}_3 \times \mathbb{Z}_3$, then $M$ must have the same 3-local homology and intersection form as $\mathbb{C P}^2$. If $G=\mathbb{Z}_q \rtimes \mathbb{Z}_3, q$ odd, $3 \nmid q$ is non-abelian, then $M$ must have the same $q$-local homology as $\mathbb{C P}^2$.
4. If $b_2(M)=0$ and $G=\mathbb{Z}_q \rtimes \mathbb{Z}_{2^r}, q$ odd, $r \geq 2$ is non-abelian, then $M$ must have the same $q$-local homology as $S^4$.

We combine these results into two main theorems: Theorem A and Theorem B in Chapter 1 of the thesis. These results strengthen the work done by Edmonds, and Hambleton and Pamuk. We remark that for $b_2(M) \leq 2$ there are other examples of finite groups which can act in the above way.
References:

1. Ian Hambleton and Semra Pamuk, Rank conditions for finite group actions on 4-manifolds, Canadian Journal of Mathematics. Journal Canadien de Math´ematiques 74 (2022), no. 2, 550–572. MR 4411001
2. Allan L. Edmonds, Homologically trivial group actions on 4-manifolds, arxiv:math/9809055 (1998).
3. Michael P. McCooey, Groups that act pseudofreely on $S^2 \times S^2$, Pacific Journal of Mathematics 230 (2007), no. 2, 381–408. MR 2309166

Venue: Zoom Link (Passcode- 2qVHVi)
Time: 7:00 pm Indian time (8:30 am Canadian time)
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We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.

Venue: 112, APC Ray LHC, IISER Kolkata
Time: 2:30-3:30 pm
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The notion of topological rigidity and Haken manifolds were introduced and several related interesting results were stated. Also, a proof sketch of Nielsen's theorem on topological rigidity of compact surfaces with boundary was presented.

Venue: 211, APC Ray LHC, IISER Kolkata
Time: 3-4 pm
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